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On the structure of left and right F-, SM-, and E-quasigroups | OMICS International
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Journal of Generalized Lie Theory and Applications
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On the structure of left and right F-, SM-, and E-quasigroups

Victor SHCHERBACOV*

Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, str. Academiei 5, MD-2028 Chisinau, Moldova
E-mail: [email protected]

Received Date: November 08, 2008; Accepted Date: January 25, 2009

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given; for example, nite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isotopic to the direct product of a group and a left S-loop (this is an answer to Belousov \1a" problem). Any left E-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(1) problem). As corollary it is obtained that any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.

Contents

1 Introduction      198

1.1 Quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

1.2 Autotopisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

1.3 Quasigroup classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

1.4 Congruences and homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 205

1.5 Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

1.6 Parastrophe invariants and isostrophisms . . . . . . . . . . . . . . . . . . . . 213

1.7 Group isotopes and identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

2 Direct decompositions       218

2.1 Left and right F-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

2.2 Left and right SM- and E-quasigroups . . . . . . . . . . . . . . . . . . . . . . 222

2.3 CML as an SM-quasigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

3 The structure      228

3.1 Simple left and right F-, E-, and SM-quasigroups . . . . . . . . . . . . . . . . 229

3.2 F-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

3.3 E-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

3.4 SM-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

3.5 Simple left FESM-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 242

4 Loop isotopes      244

4.1 Left F-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

4.2 F-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

4.3 Left SM-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

4.4 Left E-quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

1 Introduction

Murdoch introduced F-quasigroups in [77]. At this time, Sushkevich studied quasigroups with the weak associative properties [116,117]. Their name F-quasigroups obtained in an article of Belousov [8]. Later Belousov and his pupils Golovko and Florja, Ursul, Kepka, Kinyon, Phillips, Sabinin, Sbitneva, Sabinina, and many other mathematicians studied F-quasigroups and left F-quasigroups [12,15,16,28,34,35,36,41,42,53,55,86,87]. In [55,57,58] it is proved that any F-quasigroup is linear over a Moufang loop. The structure of F-quasigroups also is described in [55,57,58].

Left and right SM-quasigroups (semimedial quasigroups) are defined by Kepka. In [49] Kepka has called these quasigroups LWA-quasigroups and RWA-quasigroups, respectively. SM-quasigroups are connected with trimedial quasigroups. These quasigroup classes are studied in [6,49,50,52,63,64,106,107]. Kinyon and Phillips have defined and studied left and right E-quasigroups [64].

Main idea of this paper is to use quasigroup endomorphisms by the study of structure of quasigroups with some generalized distributive identities. This idea has been used by the study of many loop and quasigroup classes, for example, by the study of commutative Moufang loops, commutative diassociative loops, CC-loops (LK-loops), F-quasigroups, SMquasigroups, trimedial quasigroups, and so on [4,7,12,23,24,25,61,62,65,83,84]. This idea is clearly expressed in Shchukin's book [106].

Using language of identities of quasigroups with three operations in signature, i.e., of quasigroups of the form equation, we can say that we study some quasigroups from the following quasigroup classes: equationequation.

This paper is connected with the following problems.

Problem 1 (Belousov Problem 1a [12,55,98]). Find necessary and sucient conditions that a left special loop is isotopic to a left F-quasigroup.

Problem 1a has been solved partially by Florea and Ursul [34,36]. They proved that a left F-quasigroup with IP-property is isotopic to an A-loop.

Problem 2 (Problem 2.8 from [64]). (1) Characterize the loop isotopes of quasigroups satisfying (El).

(2) Characterize the loop isotopes of quasigroups satisfying (El), (Sl), and (Fl).

Problem 3. It is easy to see that in loops equation. Describe quasigroups with the property f(ab) = f(a)f(b) for all equation, where f(a) is left local identity element of a (see [99, p. 12]).

The results of this paper were presented at the conference LOOPS'07 (August 19-24, 2007, Prague). In order to make the reading of this paper more or less easy we give some necessary preliminary results and quit detailed proofs.

1.1 Quasigroups

Let equation be a groupoid (be a magma in alternative terminology). As usual, the map equationequation for all x ∈ Q, is a left translation of the groupoid equation relative to a xed element a ∈ Q; the map equation , is a right translation.

Definition 1.1. A groupoid equation is said to be a division groupoid if the mappings Lx and Rx are surjective for every x ∈ G.

In a division groupoid equation , any from equations a ⋅ x = b and y ⋅ a = b has at least one solution for any xed a, b ∈ Q, but we cannot guarantee that these solutions are unique solutions.

Definition 1.2. A groupoid equation is said to be a cancellation groupoid if equation, equation for all a, b, c ∈ G.

If any from equations a ⋅ x = b and y ⋅ a = b has a solution in a cancellation groupoid equation for some fixed a, b ∈ Q, then this solution is unique. In other words, in a cancellation groupoid, the mappings Lx and Rx are injective for every x ∈ G.

Definition 1.3. A groupoid equation is called a quasigroup if, for all a, b ∈ Q, there exist unique solutions x, y ∈ Q to the equations x ⋅ a = b and a ⋅ y = b, i.e., in this case any right and any left translation of the groupoid equation is a bijection of the set Q.

Remark 1.4. Any division cancellation groupoid is a quasigroup and vice versa.

A sub-object equation of a quasigroup equation is closed relative to the operation ⋅, i.e., if a, b ∈ H, then a ⋅ b ∈ H.

We denote by SQ the group of all bijections (permutations in nite case) of a set Q.

Definition 1.5. A groupoid (Q,A) is an isotope of a groupoid (Q,B) if there exist permutations equation of the set Q such thatequation for allequation. We also can say that a groupoid (Q,A) is an isotopic image of a groupoid (Q,B). The triple (equation) is called an isotopy (isotopism).

We will write this fact also in the form (Q,A)=(Q,B)T, where T = (equation) [12,15,83].

If only the fact will be important that binary groupoids equation and equation are isotopic, then we will use the record equation.

Definition 1.6. Isotopy of the form equation is called a principal isotopy.

Remark 1.7. Up to isomorphism any isotopy is a principal isotopy. Indeed, T = (equation) = equation.

We have the following definition of a quasigroup.

Definition 1.8 (see [14,30,76]). A binary groupoid (Q,A) such that in the equality equation knowledge of any 2 elements of x1, x2, x3 uniquely speci es the remaining one is called a binary quasigroup.

From Definition 1.8, it follows that with any quasigroup (Q,A) it is possible to associate more (3! − 1) = 5 quasigroups, the so-called parastrophes of quasigroup (Q,A):

equation

We will denote

• the operation of (12)-parastrophe of a quasigroup equation by *;

• the operation of (13)-parastrophe of a quasigroup equationby /;

• the operation of (23)-parastrophe of a quasigroup equationby \;

• the operation of (123)-parastrophe of a quasigroup equationby //;

• the operation of (132)-parastrophe of the quasigroup equationby \\;

We have defined left and right translations of a groupoid and, therefore, of a quasigroup. But for quasigroups it is possible to de ne the third kind of translations. If equation is a quasigroup, then the map equation for all x ∈ Q, is called a middle translation [13,104].

In Table 1 connections between different kinds of translations in different parastrophes of a quasigroup equation are given. This table in fact is there in [13]; see also [31,94].

equation

In Table 1, for example, equation.

If equation is an isotopy, σ is a parastrophy, then we define equationequation.

Lemma 1.9. In a quasigroup equation [12,14].

Definition 1.10. An element f(b) of a quasigroup equation is called left local identity element of an element b ∈ Q, if f(b) ⋅ b = b, in other words, f(b) = b/b.

An element e(b) of a quasigroup equation is called right local identity element of an element b ∈ Q, if b ⋅ e(b) = b, in other words, e(b) = b\b.

An element s(b) of a quasigroup equation is called middle local identity element of an element b ∈ Q, if b ⋅ b = s(b) [93,94].

An element e is a left (right) identity element for quasigroup equation which means that e = f(x) for all x ∈ Q (resp., e = e(x) for all x ∈ Q). A quasigroup with the left (right) identity element will be called a left (right) loop.

The fact that an element e is an identity element of a quasigroup equation means that e(x) = f(x) = e for all x ∈ Q, i.e., all left and right local identity elements in the quasigroup equation coincide [12].

Connections between different kinds of local identity elements in different parastrophes of a quasigroup equation are given in Table 2 [93,94].

In Table 2, for example, equation

Remark 1.11. We notice that in [6,106] the mapping s is denoted by β.

equation

Definition 1.12. A quasigroup equation with an identity element e ∈ Q is called a loop.

Quasigroup isotopy of the form equation is called an LP-isotopy. Any LP-isotopic image of a quasigroup is a loop [12,15].

Lemma 1.13 (see [15, Lemma 1.1]). Let (Q, +) be a loop and equation a quasigroup. If (Q, +) = equation , thenequation for some translations of equation.

Lemma 1.14. If equation is a quasigroup, equation is its subquasigroup, a, b ∈ H, then (H, ⋅)T is a subloop of the loop (Q, ⋅)T, where T is an isotopy of the form equation.

Proof. We have that equation are translations of equation , since a, b ∈ H.

We define the following mappings of a quasigroup equation for allequation for allequation for all x ∈ Q.

Definition 1.15 (see [12,15,21,27,32,33,83,104]). An algebra equation is called a quasigroup, if on the set Q there exist operations "\" and "/" such that in equation identities

equation

are fulfilled.

Lemma 1.16. (1) Any sub-object of a quasigroup equation is a cancellation groupoid.

(2) Any sub-object of a quasigroup equation is a subquasigroup.

(3) Any subquasigroup of a quasigroup equation is a subquasigroup in equation and, vice versa, any subquasigroup of a quasigroup equation is a subquasigroup in equation.

Proof. (1) If a, b, c ∈ H, then from a ⋅ b = a ⋅ c follows b = c, since equation. Similarly from b ⋅ a = c ⋅ a follows b = c.

(2) and (3), see [12,27,71,83].

Left, middle, and right nuclei of a loop equation are defined in the following way:

equation

Nucleus of a loop is defined in the following way: equation [12,24]. Bruck defined a center of a loop equation as equation, where

equation

Information on quasigroup nuclei can be found in [99].

1.2 Autotopisms

Definition 1.17. An autotopism (sometimes we will call autotopism an autotopy) is an isotopism of a quasigroup equation into itself, i.e., a triple (α, β, γ) of permutations of the set Q is an autotopy if the equalityequation is fulfilled for all x, y ∈ Q.

Definition 1.18. The third component of any autotopism is called a quasiautomorphism.

By Top equation we will denote the group of all autotopies of a quasigroup equation.

Theorem 1.19 (see [12,15,14]). If quasigroups equation and equation are isotopic with isotopy T, i.e., equation, thenequation.

Lemma 1.20 (see [12,15]). If equation is a loop, then any its autotopy has the formequationequation.

Proof. Let T = (α, β, γ) be an autotopy of a loop equation, i.e.,equation. If we put x = 1, then we obtainequation. If we put y = 1, then, by analogy, we obtain, equation. Thenequation, where β1 = k, α1 = d.

We can obtain more detailed information on autotopies of a group and, since autotopy groups of isotopic quasigroups are isomorphic, on autotopies of quasigroups that are some group isotopes.

Theorem 1.21 (see [15]). Any autotopy of a group (Q, +) has the form

equation

where La is a left translation of the group (Q, +), Rb is a right translation of this group, δ is an automorphism of (Q, +).

Corollary 1.22. (1) If equation, then equation. (2) Ifequation, thenequation. (3) Ifequation, then a ∈ C(Q, +).

Proof. (3) We have equation.

Corollary 1.23. Any group quasiautomorphism has the form equation, whereequation [96].

Proof. We have equation, whereequation.

Lemma 1.24. (1) If x ⋅ y = αx*y, where (Q, *) is an idempotent quasigroup, α is a permuta- tion of the set Q, then equation, in particular,equation.

(2) If x ⋅ y = x * βy, where (Q, *) is an idempotent quasigroup, β is a permutation of the set Q, then equation, in particular,equation(see [72, Corollary 12]).

Proof. (1) We give a sketch of the proof. If equation, thenequationequation. If y = αx, thenequation.

(2) The proof of Case (2) is similar to the proof of Case (1).

Quasigroup classes

Definition 1.25. A quasigroup (Q; .) is

• medial, if equation for all equation

• left distributive, if equation for all equation

• right distributive, if equation for all equation

• distributive, if it is left and right distributive;

• idempotent, if equation for all equation

• unipotent, if there exists an element equation such that equation for all equation

• left semi-symmetric, if equation for all equation

• TS-quasigroup, if equation for all equation

• left F-quasigroup, if equation for all equation

• right F-quasigroup, if equation for all equation

• left semimedial or middle F-quasigroup, if equation for all equation

• right semimedial, if equation for all equation

• F-quasigroup, if it is left and right F-quasigroup;

• left E-quasigroup, if equation for all equation

• left E-quasigroup, if equation for all equation

• right E-quasigroup, if equation for all equation

• E-quasigroup, if it is left and right E-quasigroup;

• LIP-quasigroup, if there exists a permutation λ of the set Q such that equation for all equation

• RIP-quasigroup, if there exists a permutation p of the set Q such that equation for all equation

• IP-quasigroup, if it is LIP- and RIP-quasigroup.

A quasigroup (Q; .) of the form equation whereequation is a group,equation is a permutation of the set Q, is called a left linear quasigroup; a quasigroupequation of the
form equation where equation is a group,equation is a permutation of the set Q, is called a right linear quasigroup [114, 118].

Definition 1.26. A loop equation

Bol loop (left Bol loop), ifequation for all equation

Moufang loop, ifequation for all equation

commutative Moufang loop equation if equation for all equation of the set Q,equation for allequation

right M-loop, ifequation for all equation whereequation is
a mapping of the set Q;

M-loop, if it is left M- and right M-loop;

• left special, if equation is an automorphism of equation for any pairequation

• right special, if equation is an automorphism of equation for any pairequation

In [12] the left special loop is called special. In [50,106,64] left semimedial quasigroups
are studied. A quasigroup is trimedial if and only if it is satis es left and right E-quasigroup
equality [64]. Information on properties of trimedial quasigroups is there in [63].

Every semimedial quasigroup is isotopic to a commutative Moufang loop [50]. In the
trimedial case the isotopy has a more restrictive form [50].

In a quasigroup equation the equalitiesequationequation take the formequationequation respectively, and they are identities inequation

Therefore any subquasigroup of a left F-quasigroup equation is a left F-quasigroup; any homomorphic image of a left F-quasigroup equation is a left F-quasigroup [27,71]. It is clear
that the situation is the same for right F-quasigroups, left and right E- and SM-quasigroups.

Lemma 1.27. Any medial quasigroup equation is both a left and right F-, SM-, and E-quasigroup.

Proof. Equality equation follows from medial identity equation Respectively, byequation we haveequation is a left F-quasigroup in these cases, and so on.

Lemma 1.28. (1) Any left distributive quasigroup equation is a left F-, SM-, and E-quasigroup.(2) Any right distributive quasigroup equation is a right F-, SM-, and E-quasigroup.

Proof. (1) It is easy to see that equation is idempotent quasigroup. ThereforeequationequationThen equationequation

(2) The proof of this case is similar to the proof of Case (1).

Lemma 1.29. A quasigroup equation in which

(1) the equality equation is true for all equation where δ is a map of the set Q, is a left F-quasigroup [15];

(2) the equality equation is true for all equation where δ is a map of the set Q is a right F-quasigroup;

(3) the equality equation is true for all equation where δ is a map of the set Q is a left semimedial quasigroup;

(4) the equality equation is true for all equationwhere δ is a map of the set Q is a right semimedial quasigroup;

(5) the equality equation is true for all equation is a left E-quasigroup;

(6) the equality equation is true for equation where δ is a map of the set Q ,is a right E-quasigroup.

Proof. (1) If we take equation then we haveequation Cases (2)-(6) are proved similarly.

Theorem 1.30 (Toyoda Theorem [12,15,22,78,103,119]). Any medial quasigroup equation can be presented in the form equation where equation is an Abelian group,equationare
automorphisms of equation such that equation is some xed element of the set Q and vice
versa.

Theorem 1.31 (Belousov Theorem [9,12,15]). Any distributive quasigroup equation can be
presented in the form equation where equation is a commutative Moufang loop,

equation

A left (right) F-quasigroup is isotopic to a left (right) M-loop [15,42]. A left (right) Fquasigroup
is isotopic to a left (right) special loop [8,16,12,41]. An F-quasigroup is isotopic
to a Moufang loop [55].

If a loop equation is isotopic to a left distributive quasigroup equation with isotopy the formequation then equation will be called a left S-loop. Loop equation and quasigroup equation are said to be related.

If a loop equation is isotopic to a right distributive quasigroup equation with isotopy the formequation then equation will be called a right S-loop.

Definition 1.32. An automorphism ψ of a loop equation is called complete, if there exists a permutation ψ of the set Q such that equation for all equation Permutation ψ is called a complement of automorphism ψ

The following theorem is proved in [17].

Theorem 1.33. A loop equation is a left S-loop, if and only if there exists a complete auto-
morphism
ψ of the loop equation such that at least one of the following conditions is ful lled:equation

equation for all equation Thus equation whereequation is a left distributive quasigroup which corresponds to
the loop
equation

Remark 1.34. In [17,81] a left S-loop is called an S-loop.

A left distributive quasigroup equation with identityequation is isotopic to a left Bol loop [12,15,16]. Last results of Nagy [79] let us hope on progress in researches of left distributive
quasigroups. Some properties of distributive and left distributive quasigroups are described
in [38,39,40,115].

Theorem 1.35. Any loop which is isotopic to a left F-quasigroup is a left M-loop (see [15, Theorem 3.17, p. 109]).

Theorem 1.36 (Generalized Albert Theorem). Any loop isotopic to a group is a group [2,3,12,15,68,83,99].

Congruences and homomorphisms

Results of this subsection are standard, well-known [12,24,83,71,27], and slightly adapted
for our aims.

A binary relation ψ on a set Q is a subset of the cartesian product equation [21,71,82].

If ψ and ψ are binary relations on Q, then their product is de ned in the following way:equation if there is an element equation such thatequation and equation The last
condition is written also in such form equation

Theorem 1.37. Let S be a nonempty set and let equation be a relation between elements of S that
satis es the following properties:

equation

equation

equation

Then equation yields a natural partition of S, whereequation is the cell containing a for allequation Conversely, each partition of S gives rise to a natural relation equation satisfying the re
exive, symmetric, and transitive properties if equation is de ned to mean thatequation

Definition 1.38. A relation equation on a set S satisfying the re
exive, symmetric, and transitive
properties is called an equivalence relation on S. Each cell equation in the natural partition given
by an equivalence relation is an equivalence class.

Definition 1.39. An equivalenceequation is a congruence of a groupoid equation if the following implications are true for allequation

In other words, equivalence equation is a congruence of equation if and only if equation is a subalgebra ofequation Therefore we can formulate Definition 1.39 in the following form.

Definition 1.40 (see [29]). An equivalence equation is a congruence of a groupoid equation if the
following implication is true for all equation

Definition 1.41 (see [12,15]). A congruence equation of a quasigroupequation is normal, if the following implications are true for allequation

Definition 1.42. An equivalence equation is a congruence of a quasigroupequation if the following
implications are true for all equation

equation

equation

equation

One from the most important properties of e-quasigroup equation is the following property.

Lemma 1.43 (see [12,15,21,71]). Any congruence of a quasigroup equation is a normal congruence of quasigroup equation any normal congruence of a quasigroup equation of quasigroup equation

Denition 1.44. If equation is a binary relation on a set equation is a permutation of the set Q
and from equation it followsequation and equation for all equation then we will say that the permutation is an admissible permutation relative to the binary relation equation [12,100].

Moreover, we will say that a binary relation equation admits a permutation α

Lemma 1.45 (see [13]). Any normal quasigroup congruence is admissible relative to any
left, right, and middle quasigroup translation.

Proof. The fact that any normal quasigroup congruence is admissible relative to any left
and right quasigroup translation follows from Definitions 1.39 and 1.41.

Let equation be a normal congruence of a quasigroup equation Prove the following implication:

equation (1.5)

If equationequation Similarly if equation then equationequation Since equation is a congruence of quasigroup equation (Lemma 1.43), then implication
(1.5) is true.

Implication

equation

is proved in a similar way. If equation then equation Similarly if equation then equation Since equation is a congruence of quasigroup equation (Lemma 1.43), then implication (1.6) is true.

Corollary 1.46. If equation is a normal quasigroup congruence of a quasigroup Q, then equation is a nor-
mal congruence of any parastrophe of Q [13].

Proof. The proof follows from Lemma 1.45 and Table 1.

In Lemma 1.48 we will use the following fact about quasigroup translations and normal
quasigroup congruences.

Lemma 1.47. If equation

Proof. If equation and equation then equation Indeed, ifequation then equation if equation and equation then equation and, nally,equation In other words, ifequation and equation then equation

Since equation and equation is a normal quasigroup congruence, we haveequationequation

We give a sketched proof of the following well-known fact [27,70,108]. We follow [108].

Lemma 1.48. Normal quasigroup congruences commute in pairs.

Proof. Let equation and equation be normal congruences of a quasigroup equation Then equation means that there exists an elementequation such thatequation and equation

Further, we have

equation

equation

equation

Then

equation

From relations

equation

equation

equation

we obtain

equation

Therefore, equation

See [37] for additional information on permutability of quasigroup congruences.

Definition 1.49 (see [83]). If equationand equation are binary quasigroups, h is a single-valued mapping of Q into H such thatequation then h is called a homomorphism (a multiplicative homomorphism) of equation into equation and the setequation is called
homomorphic image of
equation under h.

In case equation a homomorphism is also called an endomorphism, and an isomorphism
is referred to as an automorphism.

Lemma 1.50. (1) Any homomorphic image of a quasigroup equation is a division groupoid [5,24].

(2) Any homomorphic image of a quasigroup equation is a quasigroup [27,71].

Proof. (1) Let equation We demonstrate that solution of equationequation lies inequation Consider the equationequation Denote solution of this equation by c. Then h(c) is solution to the equationequationIndeed equation ). For
equation equation the proof is similar.

(2) see [12,15,27,71,83].

Let h be a homomorphism of a quasigroup equation onto a groupoid equation Then h induces a congruence Kerequation(the kernel of h) in the following way, equation if and only ifequation [15,83].

Theorem 1.51 (see [15],[83, Theorem I.7.2]). If h is a homomorphism of a quasigroupequation onto a quasigroupequation then h determines a normal congruenceequation such that equationequation and vice versa, a normal congruence  induces a homomorphism from equation ontoequation

A subquasigroup equation of a quasigroup equation is normalequation ifequation is
an equivalence class (in other words, a coset class) of a normal congruence.

Lemma 1.52. An equivalence class equation of a congruence equation of a quasigroup equation is
a sub-object of equation if and only ifequation

Proof. We recall by Lemma 1.16 any quasigroup sub-object is a cancellation groupoid. The
proof is similar to the proof of Lemma 1.9 from [15]. If equation and equation then equation moreover equation since equation Conversely, letequation If equation then equation and equation Thenequation

Lemma 1.53 (see [12], [15, Lemma 1.9]). An equivalence class equation of a normal congruence equation of a quasigroup equation is a subquasigroup of equation if and only ifequation

Lemma 1.54. If h is an endomorphism of a quasigroup equation then equation is a subquasigroup
of equation

Proof. We rewrite the proof from [15, p. 33] for slightly more general case. Prove that equation is a subquasigroup of quasigroup equation.Let equation We demonstrate that solution of equationequation lies in equation Consider the equation equation Denote
solution of this equation by c. Then h(c) is solution of equation equationIndeed, equation

It is easy to see that this is a unique solution. Indeed, if equation then equationequation Since equation are elements of quasigroup equation then equation

For equation equation the proof is similar.

Remark 1.55. It is possible to give the following proof of Lemma 1.54. The equation is
a cancellation groupoid, since it is a sub-object of the quasigroup equation (Lemma 1.16). From the other side equation is a division groupoid, since it is a homomorphic image of equation (Lemma 1.50). Therefore by Remark 1.4 equation is a subquasigroup of the quasigroup equation

Corollary 1.56. (1) Any subquasigroup equation of a left F-quasigroup equation is a left F- quasigroup.

(2) Any endomorphic image of a left F-quasigroup equation is a left F-quasigroup.

Proof. (1) If equation then the solution of equation equation also is in H.

(2) From Case (1) and Lemma 1.54 it follows that any endomorphic image of a left
F-quasigroup equation is a left F-quasigroup.

Remark 1.57. The same situation is for right F-quasigroups, left and right E-, and SMquasigroups
and all combinations of these properties.

Corollary 1.58. If h is an endomorphism of a quasigroup equation then h is an endomorphism of the quasigroupsequation i.e., fromequation we
obtain that

equation

equation

equation

equation

equation

Proof. From Lemma 1.54 we have that equation is a subquasigroup ofequation

(1) If we pass from the quasigroup equation to quasigroupequation then subquasigroupequation of the quasigroup equation will correspond to the subquasigroup equation Indeed, any subquasigroup
of the quasigroup equation is closed relative to parastrophe operationsequation of the
quasigroup equation Further we haveequation

(2) If we pass from the quasigroup equation to quasigroupequation then subquasigroupequation of the quasigroup equation will correspond to the subquasigroupequation

Let equation where equation Then from de nition of the operation / it follows that equationequation (see [71, p. 96, Theorem 1]). The remaining cases are proved in the similar way.

Lemma 1.59 (see [12,102]). If equation is a nite quasigroup, then any of its congruences is
normal, any of its homomorphic images is a quasigroup.

Lemma 1.60. Let equation be a quasigroup.

• If f is an endomorphism of equation thenequation for allequation

• If e is an endomorphism of equation thenequation for allequation

• If s is an endomorphism of equation equation ([64, Lemma 2.4.]).

Proof. We will use Corollary 1.58.

• If f is an endomorphism, then equationequation

• If e is an endomorphism, then equation equation

• If s is an endomorphism, then equationequation

This proves the lemma.

The group equation is a quasigroup, is called multiplication group of quasigroup.

The group equation is called inner mapping group of a quasigroup (Q, .) relative to an element equation Group equation is stabilizer of a fixed element h by action equation on the set Q. In loop case usually it is studied the group equation here 1 is the identity element of a loop (Q,. ).

Theorem 1.61. A subquasigroup H of a quasigroup Q is normal if and only if equation for a fixed element equation [12].

In [12, p. 59] the following key lemma is proved.

Lemma 1.62. Let θ be a normal congruence of a quasigroup (Q,. ). If a quasigroup (Q; ο) is isotopic to (Q,. ) and the isotopy equation is admissible relative to θ, then θ is a normal congruence also in (Q; ο).

For our aims we will use the following theorem.

Theorem 1.63 (see [80, 60, 94, 96]). Let (Q, +) be an IP-loop, equation where equation be a normal congruence of (Q, +). Then θ is normal congruence of (Q,.) if and only if equation are automorphisms of Ker θ.

We denote by nCon(Q,.) the set of all normal congruences of a quasigroup (Q,.).

Corollary 1.64. If (Q,.) is a quasigroup, (Q, +) is a loop of the form equationfor all equation then equation

Proof. If θ is a normal congruence of a quasigroup (Q,.), then, since θ is admissible relative to the isotopy equation θ is also a normal congruence of a loop (Q, +).

In loop case situation with normality of subloops is well known and more near to the group case [24, 83, 12, 15]. As usual a subloop (H, +) of a loop (Q, +) is normal, if H = θ(0) = Ker θ, where θ(0) is an equivalence class of a normal congruence θ that contains identity element of (Q, +) [12, 83]. We will name congruence θ and subloop (H, +) by corresponding.

Example 1.65. In the group equation there exists endomorphism equation and equation

Example 1.66. In the cyclic group (Z4, +), Z4 = {0; 1; 2; 3}, there exists endomorphism equation such that equation The endomorphism h defines normal congruence θ with the following coset classes: θ(0) = {0; 2} and θ(1) = {1; 3}. It is clear that equation

Definition 1.67. A normal subloop (H; +) of a loop (Q, +) is admissible relative to a permutation α of the set Q if and only if the corresponding to (H; +) normal congruence θ is admissible relative to α.

Definition 1.68. A quasigroup (Q,.) is simple if its only normal congruences are the diagonal equation

Definition 1.69. We will name a subloop (H, +) of a loop (Q, +)α-invariant relative to a permutation α of the set Q, if αH = H.

We will name a loop (Q, +)α-simple if only identity subloop and the loop (Q, +) are invariant relative to the permutation α of the set Q.

We will name a quasigroup (Q,.)α-simple relative to the permutation α of the set Q, if only the diagonal and universal congruences are admissible relative to α.

Corollary 1.70. Let (Q,.) = (Q, +)equation, where (Q, +) is a loop, αequationIf (Q, +) does not contain normal subloops admissible relative to permutations α; fi, then quasigroup (Q,.) is simple.

Proof. The proof follows from Lemmas 1.62 and 1.13 and Corollary 1.64.

Direct products

Definition 1.71. If (Q1,.), (Q2,. ο) are binary quasigroups, then their (external) direct prod- uct equation is the set of all ordered pairs equation and where the operation in (Q,. *) is defined componentwise, that is, equation

Direct product of quasigroups is studied in many articles and books; see, for example, [18, 19, 29, 45, 108, 80]. The concept of direct product of quasigroups was used already in [78]. In group case it is possible to find these definitions, for example, in [44].

In [27, 108, 109] there is a definition of the (internal) direct product of Ω-algebras. We recall that any quasigroup is an Ω-algebra.

Let U and W be equivalence relations on a set A, equation equation is an equivalence relation on A called the join of U and W. If U and W are equivalence relations on A for which equation are said to commute [108].

If Ω is an -algebra and U, W are congruences on A, then equation, and equation are also congruences on A.

Definition 1.72 (see [108, 109]). If U and W are congruences on the algebra A which commute and for which equation then the join equation and W is called direct product equation of U and W.

The following theorem establishes the connection between concepts of internal and external direct products of Ω-algebras.

Theorem 1.73 (see [108, p. 16], [109]). An Ω-algebra A is isomorphic to a direct product of Ω-algebras B and C with isomorphism equation if and only if there exist such congruences U and W of A that equation.

We will use the following easy proved fact.

Lemma 1.74. If a loop Q is isomorphic to the direct product of the loops A and B, then equation

Lemma 1.75. If a left F-quasigroup Q is isomorphic to the direct product of a left F- quasigroup A and a quasigroup B, then B also is a left F-quasigroup.

Proof. Indeed, if q = (a; b), where equation then e(q) = (e(a); e(b)).

Remark 1.76. An analog of Lemma 1.75 is true for right F-quasigroups, left and right SMand E-quasigroups.

There exist various approaches to the concept of semidirect product of quasigroups [89, 88, 26, 120]. By an analogy with group case [47] we give the following definition of the semidirect product of quasigroups. Main principe is that a semidirect product is a cartesian product as a set [120].

Definition 1.77 (see [121]). Let Q be a quasigroup, A a normal subquasigroup of Q (i.e., equation and B a subquasigroup of Q. A quasigroup Q is the semidirect product of quasigroups A and B, if there exists a homomorphism equation which is the identity on B and whose kernel is A, i.e., A is a coset class of the normal congruence Ker h. We will denote this fact as follows: equation

Remark 1.78. From results of Mal'tsev [70], see, also, [100], it follows that normal subquasigroup A is a coset class of only one normal congruence of the quasigroup Q.

Lemma 1.79. If a quasigroup Q is the semidirect product of quasigroups A and B, equation then there exists an isotopy T of Q such that QT is a loop and equation

Proof. If we take isotopy of the form equation, where a equation A, then we have that QT is a loop, AT is its normal subloop (Lemma 1.62, Remark 1.45). Further we have that BT is a loop since equation Therefore BT is a subloop of the loop QT, since the set B is a subset of the set Q.

Corollary 1.80. If a quasigroup Q is the direct product of quasigroups A and B, then there exists an isotopy T = (T1; T2) of Q such that equation is a loop.

Proof. The proof follows from Lemma 1.79.

Lemma 1.81. (1) If a linear left loop (Q,.) with the form equation is a group, 2 Aut(Q, +), is the semidirect product of a normal subgroup equation and a subgroup equation then (Q,.) = (Q, +).

(2) If a linear right loop (Q,.) with the form equation where (Q, +) is a group, equation is the semidirect product of a normal subgroup equation and a subgroup equation

Proof. (1) Since (Q,.) is the semidirect product of a normal subgroup (H; ) and a subgroup (K; ), then we can write any element a of the loop (Q,.) in a unique way as a pair a = (k; 0)  (0; h), whereequation We noticeequation since (K; ), (H; ) are subgroups of the left loop (Q,.). Indeed, fromequation for allequation we haveequation

Further we have equationequation

(2) This case is proved similarly to Case (1).

Example 1.82. Medial quasigroup equation where (Z9; +)equation is the cyclic group, equation demonstrates that some restrictions in Lemma 1.81 are essential.

Parastrophe invariants and isostrophisms

Parastrophe invariants and isostrophisms are studied in [13].

Lemma 1.83. If a quasigroup Q is the direct product of a quasigroup A and a quasigroup B, then equation, where σ is a parastrophy.

Proof. From Theorem 1.73 it follows that the direct product A  B defines two quasigroup congruences. From Theorem 1.51 it follows that these congruences are normal. By Corollary 1.46 these congruences are invariant relative to any parastrophy of the quasigroup Q.

Lemma 1.84. If Q is a quasigroup and α equation, then α equation where σ is a paras- trophy.

Proof. It is easy to check [93, 94].

Lemma 1.85. (1) A quasigroup (Q,.) is a left F-quasigroup if and only if its (12)-parastro- phe is a right F-quasigroup.

(2) A quasigroup (Q,.) is a left E-quasigroup if and only if its (12)-parastrophe is a right E-quasigroup.

(3) A quasigroup (Q,.) is a left SM-quasigroup if and only if its (12)-parastrophe is a right SM-quasigroup.

(4) A quasigroup (Q,.) is a left distributive quasigroup if and only if its (12)-parastrophe is a right distributive quasigroup.

(5) A quasigroup (Q,.) is a left distributive quasigroup if and only if its (23)-parastrophe is a left distributive quasigroup.

(6) A quasigroup (Q,.) is a left SM-quasigroup if and only if (Q; \) is a left F-quasigroup.

(7) A quasigroup (Q,.) is a right SM-quasigroup if and only if (Q; /) is a right F- quasigroup.

(8) A quasigroup (Q,.) is a left E-quasigroup if and only if (Q; \) is a left E-quasigroup (see [64, Lemma 2.2]).

(9) A quasigroup (Q,.) is a right E-quasigroup if and only if (Q; /) is a right E-quasigroup (see [64, Lemma 2.2]).

Proof. It is easy to check Cases (1){(4).

(5) The fulfilment in a quasigroup (Q,.) of the left distributive identity is equivalent to the fact that in this quasigroup any left translation equation is an automorphism of this quasigroup. Indeed, we can rewrite left distributive identity in such manner equation Using Table 1 we have that equation Thus by Lemma 1.84 equation Therefore, if (Q,.) is a left distributive quasigroup, then (Q; \) also is a left distributive quasigroup and vice versa.

(6) Let (Q; n) be a left F-quasigroup. Then

equation

If equation We notice, ifequation thenequation See Table 2.

We can rewrite equality equationequation Now we have the equality equation If we denoteequation thenequation

Therefore we can rewrite equality equation in the form equationequation i.e., in the form equation

In a similar way it is possible to check the converse: if (Q,. ) is a left SM-quasigroup, then (Q,.) is a left F-quasigroup.

Cases (7)-(9) are proved in a similar way.

Corollary 1.86. If (Q,.) is a group, then

(1) (Q; n) is a left SM-quasigroup;

(2) (Q; =) is a right SM-quasigroup.

Proof. (1) Any group is a left F-quasigroup since in this case e(x) = 1 for all equation Therefore we can use Lemma 1.85(6).

(2) We can use Lemma 1.85(7).

Definition 1.87 (see [14]). A quasigroup (Q;B) is an isostrophic image of a quasigroup (Q;A) if there exists a collection of permutations equation equation are permutations of the set Q such that

equation

for all equation

A collection of permutations equation will be called an isostrophism or an isostrophy of a quasigroup (Q;A). We can rewrite equality from Definition 1.87 in the form equation

Lemma 1.88 (see [14]). An isostrophic image of a quasigroup is a quasigroup.

Proof. The proof follows from the fact that any parastrophic image of a quasigroup is a quasigroup and any isotopic image of a quasigroup is a quasigroup.

From Lemma 1.88 it follows that it is possible to define the multiplication of isostrophies of a quasigroup operation defined on a set Q.

Definition 1.89. If equation are isostrophisms of a quasigroup (Q;A), then

equation

where equation for any quasigroup triplet equation [105].

Slightly other operation on the set of all isostrophies (multiplication of quasigroup isostrophies) is defined in [14]. Definition from [69] is very close to Definition 1.89. See, also, [13, 48].

Corollary 1.90. One hasequation

Lemma 1.91. One hasequation

Proof. Let equation be an isotopy of a quasigroupequationequation Then

equation

Group isotopes and identities

Information for this subsection has been taken from [1, 10, 11, 14, 67, 114, 118].We formulate famous Four quasigroups theorem [1, 10, 14, 114] as follows.

Theorem 1.92. A quadruple equation of binary quasigroup operation defined on a non- empty set Q is the general solution of the generalized associativity equation

equation

if and only if there exists a group (Q, +) and permutations equation of the set Q such that equation

Lemma 1.93 (see Belousov criteria [11]). If in a group (Q, +) the equality equation holds for all equation where equation are some fixed permutations of Q, then (Q, +) is an Abelian group.

There exists also the following corollary adapted for our aims from results of Sokhatskii (see [114, Theorem 6.7.2]).

Corollary 1.94. If in a principal group isotope (Q,.) of a group (Q, +) the equality equationequation holds for all equation where equation are some fixed permutations of Q, then (Q, +) is an Abelian group.

Proof. If equation then we can rewrite the equality equation in the form equation Now we can apply the Belousov criteria (Lemma 1.93).

Lemma 1.95. (1) For any principal group isotope (Q,.) there exists its form equation such that α 0 = 0 [111].

(2) For any principal group isotope (Q,.) there exists its form equation such that β 0 = 0.

(3) For any right linear quasigroup (Q,.) there exists its form equation such that α0 = 0.

(4) For any left linear quasigroup (Q,.) there exists its formequation such that β0 = 0.

(5) For any left linear quasigroup (Q,.) with idempotent element 0 there exists its form equation such that β0 = 0.

(6) For any right linear quasigroup (Q,.) with idempotent element 0 there exists its form equation such that α0 = 0.

Proof. (1) We have equation

(2) We have equation

(3) We have equation where equation Since equation is an inner automorphism of the group (Q, +), we obtain equation

(4)We have equation, where equation.

(5) If equation, then equation. Therefore equation and equation.

(6) If equation, then equation. Therefore equation and equation.

Moreover, equation is an automorphism of (Q, +) as the product of two automorphisms of the group (Q, +).

Lemma 1.96. For any left linear quasigroup equation there exists its form such that equationequation.

For any right linear quasigroup equation there exists its form such that equation.

Proof. We can rewrite the form equation of a left linear quasigroup equation as follows: equation, where equation.

We can rewrite the form equation of a right linear quasigroup equation as follows: equation, where equation

Classical criteria of a linearity of a quasigroup are given by Belousov in [11]. We give a partial case of Sokhatskii result (see [112], [113, Theorem 3], [114, Theorem 6.8.6]).

We recall that up to isomorphism every isotope is principal (Remark 1.7).

Theorem 1.97. Let equation be a principal isotope of a group (Q, +), equation.

If equation is true for all equation, where equation are permutations of the set Q, a is a xed element of the set Q, then equationis a left linear quasigroup.

If equation is true for all equation, where equationare permutations of the set Q, a is a xed element of the set Q, then equation is a right linear quasigroup.

Proof. We follow [114]. By Lemma 1.95 quasigroup equation can have the form equation over a group (Q, +) such that equation. If we pass in the equalityequation to the operation "+", then we obtain equationequation

Then the permutation α is a group quasiautomorphism. It is known that any group quasiautomorphism has the form equation, where equation. See [15,12] or Corollary 1.23. Therefore equation, since equation.

By Lemma 1.95 there exists the form equation of quasigroup equation such that equation. If we pass in the equality equationto the operation "+", then we obtain equation.

Then the permutation β is a group quasiautomorphism. Therefore equation, since equation.

Corollary 1.98. (1) If a left F-quasigroup (E-quasigroup, SM-quasigroup) is a group isotope, then this quasigroup is right linear.

(2) If a right F-quasigroup (E-quasigroup, SM-quasigroup) is a group isotope, then this quasigroup is left linear [113].

Proof. The proof follows from Theorem 1.97.

Lemma 1.99. (1) If in a right linear quasigroup equationover a group (Q, +) the equality equation holds for all equation and fixed equation, then (Q, +) is an Abelian group.

(2) If in a left linear quasigroup equation over a group (Q, +) the equality equationholds for all equation and fixed equation, then (Q, +) is an Abelian group.

Proof. (1) By Lemma 1.96 we can take the following form of equation: equation. Thus we have equation, equation. Therefore α is a quasiautomorphism of the group (Q, +). Let equation, where equation.

Further we have equationequation. Finally, we can apply Lemma 1.93.

Case (2) is proved in a similar way.

Quasigroup equation with equality equation for all equation is called "quasigroup which ful lls Sushkevich postulate A".

Quasigroup equation with equality equation for all equation will be called "quasigroup which ful lls Sushkevich postulate A*".

Theorem 1.100. (1) If quasigroup equation fulfills Sushkevich postulate A, then equation is iso- topic to the group equation(see [15, Theorem 1.7]).

(2) If quasigroup equation fulfills Sushkevich postulate equation, then equation is isotopic to the group equation.

Proof. Case (1) is proved in [15].

The proof of Case (2) is similar to the proof of Case (1). It is easy to see that equation is quasigroup. Indeed, if z = c, then we have equation is isotope of quasigroup equation. Therefore equation is a quasigroup. Moreover, equation, where equation.

Quasigroup equation is a group. It is possible to use Theorem 1.92 but we give direct proof similar to the proof from [15]. We have equationequation.

Quasigroup equation with generalized identity equation, where equationis a fixed permutation of the set Q, is called "quasigroup which fulfills Sushkevich postulate B".

Quasigroup equation with generalized identity equation, where equation is a fixed permutation of the set Q, will be called "quasigroup which fulfills Sushkevich postulate B*".

It is easy to see that any quasigroup with postulate B (B*) is a quasigroup with postulate A (A*).

Theorem 1.101. (1) If quasigroup equation fulfills Sushkevich postulate B, then equation is iso- topic to the group equation, where equation (see [15, Theorem 1.8]).

(2) If quasigroup equation fulfills Sushkevich postulate B*, then equation is isotopic to the group equation, where equation.

Proof. Case (1) is proved in [15]. It is easy to see that quasigroup equationhas the right identity element, i.e., equation is right loop. Indeed, equation for all equation, where 0 is zero of group equation.

(2) The proof of Case (2) is similar to the proof of Case (1). Here we give the direct proof because the book [15] is rare. Since the quasigroup equation fulfills postulates A* and B*, then by Theorem 1.100(2), groupoid (magma)equation, is a group and equationequation. By the same theorem equation. Therefore equation is an autotopy of the group equation. By Corollary 1.22 equation. Therefore equation. It is easy to see that equation is left loop.

2 Direct decompositions

2.1 Left and right F-quasigroups

In order to study the structure of left F-quasigroups we will use approach from [78,102]. As usualequation, and so on.

Lemma 2.1 (see [77,15]). (1) In a left F-quasigroup equation the map ei is an endomorphism of equation a subquasigroup of quasigroup equation for all suitable values of the index i.

(2) In a right F-quasigroup equation the map fi is an endomorphism of equation is a subquasigroup of quasigroup equation for all suitable values of the index i

Proof. (1) From identity equationwe have equation, i.e., equation. Further we have equation and so on. Therefore equation is an endomorphism of the quasigroup equation. The fact that equation is a subquasigroup of quasigroup equation follows from Lemma 1.54.

(2) The proof is similar.

The proof of the following lemma has taken from [15, p. 33].

Lemma 2.2. (1) Endomorphism e of a left F-quasigroup equation is zero endomorphism, i.e., equation for all equation, if and only if left F-quasigroup equation is a right loop, isotope of a group (Q, +) of the form equation, whereequation.

(2) Endomorphism f of a right F-quasigroup equation is zero endomorphism, i.e., f(x) = k for all equation, if and only if right F-quasigroup equation is a left loop, isotope of a group (Q, +) of the form equation, where equation.

Proof. (1) We can rewrite equality equation in the form equation, where equation. Therefore Sushkevich postulate B is fulfilled in equationand we can apply Theorem 1.101. Further we have equation. From the other side equation. Therefore, k = 0. It is easy to see that the converse also is true.

(2) We can use the "mirror" principles.

Lemma 2.3 (see [15]). (1) The endomorphism e of a left F-quasigroup equation is a permutation of the set Q if and only if quasigroup equation of the form equation is a left distributive quasigroup and equation.

(2) The endomorphism f of a right F-quasigroup equation is a permutation of the set Q if and only if quasigroup equation of the form equation is a right distributive quasigroup and equation.

Proof. (1) Prove that equation is left distributive. We have

equation (2.1)

Prove that equation. We have equation [72].

Conversely, let equation be an isotope of the form equation, where equation, a left distributive quasigroup equation. The fact that equation follows from Lemma 1.24.

We can use equalities (2.1) by the proving that equation is a left F-quasigroup. The fact that equation follows from Lemma 1.29.

(2) The proof is similar.

In a left F-quasigroup equation define the following (maybe infinite) chain:

equation (2.2)

 

Definition 2.4. Chain (2.2) becomes stable means that there exists a number m (finite or infinite) such that equation We notice, in other words

equation

In this case we will say that endomorphism e has the order m.

Lemma 2.5. In any left F-quasigroup Q chain (2.2) becomes stable, i.e., the map equation is an automorphism of quasigroup equation.

Proof. We have two cases. (1) Chain (2.2) becomes stable on a nite step m. It is clear that in this case equation is an automorphism of equation.

(2) Prove that chain (2.2) will be stabilized on the step equation, if it is not stabilized on a finite step m. Denote equation by C.

Notice, if equation, then equation. Indeed, if equation, then equation. If equation, then equation and equation. Therefore equation and equation. Then

equation

Prove that equation. Any element equation has the form equation, where equation. Then equation for any equation. Therefore there does not exist element x of the set C such thatequation.

Therefore for any m (finite or infinite) equation is an automorphism equation.

Example 2.6. Quasigroup equation, where equation is in nite cyclic group, is medial, unipotent, left F-quasigroup such that equation. Notice in this case equation. In [71, p. 59] a mapping similar to the mapping e is called isomorphism and the embedding of an algebra in its subalgebra.

Theorem 2.7. (1) Any left F-quasigroup equation has the following structure:

equation

where equation is a quasigroup with a unique idempotent element; equation is isotope of a left distributive quasigroup equation for all equation.

(2) Any right F-quasigroup equationhas the following structure:

equation

where equation is a quasigroup with a unique idempotent element; equation is isotope of a right distributive quasigroup equationequation.

Proof. The proof of this theorem mainly repeats the proof of Theorem 6 from [102].

If the map e is a permutation of the set Q, then by Lemma 2.3 equation is isotope of left distributive quasigroup.

If equation, where k is a fixed element of the set Q, then the quasigroup equation is a quasigroup with right identity element k, i.e., it is a right loop, which is isotopic to a group (Q, +) (Lemma 2.2).

Let us suppose that equation, where m > 1.

From Lemma 2.5 it follows that equation is a subquasigroup of quasigroup equation It is clear that equation is a left F-quasigroup in which the mapequation is a permutation of the set equation. In other words, equation.

Define binary relation equation on quasigroup equation by the following rule: equation if and only if equation. Define binary relation equation on quasigroup equation by the rule equation if and only if equation, i.e., for any equation there exists exactly one element equation such that equation and, vice versa, for any equation there exists exactly one element equation such that equation.

From Theorem 1.51 and Lemma 1.54 it follows that equation is a normal congruence.

It is easy to check that binary relation equation is equivalence relation (see Theorem 1.37).

We prove that binary relation equation is a congruence, i.e., that the following implication is true:equation.

Using the definition of relation equation we can rewrite the last implication in the following equivalent form: if

equation (2.3)

then equation.

If we multiply both sides of equalities (2.3), respectively, then we obtain the following equality:

equation

Using left F-quasigroup equality equation from the right to the left and, taking into consideration that if equation, then equation, we can rewrite the last equality in the following form:

equation

since equation is a subquasigroup and, therefore, equation. Thus the binary relation equation is a congruence.

Prove that equation. From re exivity of relations equation, equation it follows that equation.

Let equation, i.e., let equation and equation where equation. Using the definitions of relations equation, equation we have equation and equation. Then there exist equation such that equation. Applying to both sides of last equality the map equation we obtain equation, since the map equation is a permutation of the set B. If a = b, then from equality equation we obtain x = y.

Prove that equation. Let a, c be any fixed elements of the set Q. We prove the equality if it will be shown that there exists element equation such that equation and equation.

From definition of congruence equation we have that condition equation is equivalent to equality equation. From definition of congruence equation it follows that condition equation is equivalent to the following condition: equation.

We prove the equality if it will be shown that there exists element equation such that equation. Such element y there exists since equation.

Prove that equation. Let a, c be any fixed elements of the set Q. We prove the equality if it will be shown that there exists element equation such that equation and equation.

From definition of congruence equation we have that condition equation is equivalent to equality equation. From definition of congruence equation it follows that condition equation is equivalent to the following condition: equation.

We prove the equality if it will be shown that there exists element equation such that equation. Such element y there exists since equation.

Therefore equation and we can use Theorem 1.73. Now we can say that quasigroup equation is isomorphic to the direct product of a quasigroup equation (Theorem 1.51) and a division groupoid equation [5,24].

From Definition 1.71 it follows, if equation, where equation, equationare quasigroups, then equation also is a quasigroup. Then by Theorem 1.51 the congruence equation is normal,equation.

Left F-quasigroup equality holds in quasigroup equation since equation.

If the quasigroups equation and equation are left F-quasigroups, equation, then equationalso is a left F-quasigroup (Lemma 1.75).

Prove that the quasigroup equation, where equation, has a unique idempotent element.

We can identify elements of quasigroup equation with cosets of the form equation, where equation.

From properties of quasigroup equation we have that equation, where the element a is a fixed element of the set A that corresponds to the coset class B. Further, taking into consideration the properties of endomorphism e of the quasigroup equation, we obtain equationequation. Therefore equation, i.e., the element a is an idempotent element of quasigroup equation.

Prove that there exists exactly one idempotent element in quasigroup equation. Suppose that there exists an element c of the set A such that equation, i.e., such that equation. Then we have equation, since equation.

The fact that equation is isotope of a left distributive quasigroup equation follows from Lemma 2.3.

Properties of right F-quasigroups coincide with the \mirror" properties of left F-quasigroups.

We notice, in finite case all congruences are normal and permutable (Lemmas 1.59 and 1.48). Therefore for finite case Theorem 2.7 can be proved in more short way.

We add some details on the structure of left F-quasigroup equation. By equation we denote endomorphic image of the quasigroup equation relative to the endomorphism ej.

Corollary 2.8. If equation is a left F-quasigroup, then equation.

Proof. This follows from the fact that the binary relation equation from Theorem 2.7 is a normal congruence in equation and subquasigroup equation is an equivalence class of equation.

Remark 2.9. For brevity we will denote the endomorphism equation such that

equation

by ej , the endomorphism equation by fj , the endomorphism equation by sj .

Corollary 2.10. If (Q, .) is a left F-quasigroup with an idempotent element, then equivalence class (cell) equation of the normal congruence Ker ej containing an idempotent element a equation Q forms linear right loop for all suitable values of j.

Proof. By Lemma 1.53 (equation, . ) is a quasigroup. From properties of the endomorphism e we have that in (equation, . ) endomorphism e is zero endomorphism. Therefore in this case we can apply Lemma 2.2. Then (equation, . ) is isotopic to a group with isotopy of the form equation , where Aut(equation, . ).

Corollary 2.11. If (Q, . ) is a right F-quasigroup, then equation.

Proof. The proof is similar to the proof of Corollary 2.8.

Corollary 2.12. If (Q, . ) is a right F-quasigroup with an idempotent element, then equiv- alence class equation of the normal congruence Ker fj containing an idempotent element a equation Q forms linear left loop (equation; ) for all suitable values of j.

Proof. The proof is similar to the proof of Corollary 2.10.

Left and right SM- and E-quasigroups

We can formulate theorem on the structure of left semimedial quasigroup using connections between a quasigroup and its (23)-parastrophe (Lemma 1.85), but in order to have more information about left semimedial quasigroup we prefer to give direct formulations some results from Section 2.1.

Lemma 2.13. (1) In a left semimedial quasigroup (Q, . ) the map si is an endomorphism of (Q, . ), si(Q, . ) is a subquasigroup of quasigroup (Q, . ) for all suitable values of the index i [77, 15].

(2) In a right semimedial quasigroup (Q, . ) the map si is an endomorphism of (Q, . ), si(Q, . ) is a subquasigroup of quasigroup (Q, . ) for all suitable values of the index i.

(3) In a left E-quasigroup (Q, . ) the map fi is an endomorphism of (Q, . ), fi(Q, . ) is a subquasigroup of quasigroup (Q, . ) for all suitable values of the index m [64].

(4) In a right E-quasigroup (Q, . ) the map ei is an endomorphism of (Q, . ), ei(Q, . ) is a subquasigroup of quasigroup (Q, . ) for all suitable values of the index m [64].

Proof. (1) From identity xx.yz = xy.xz by z = y we have xx.yy = xy.xy, i.e., s(x).(y) = s(x.y). Therefore si is an endomorphism of the quasigroup (Q, . ).

The fact that si(Q, . ) is a subquasigroup of quasigroup (Q, . ) follows from Lemma 1.54.

(2) The proof of Case (2) is similar to the proof of Case (1), thus we omit it.

(3) From identity x.yz = f(x)y.xz by y = f(y), z = y we have xy = f(x)f(y).xy. But f(xy).xy = xy. Therefore f(x). f(y) = f(x.y) [64].

(4) From identity zy.x = zx.ye(x) by y = e(y), z = y we have yx = yx.e(y)e(x).

Theorem 2.14. (1) If the endomorphism s of a left semimedial quasigroup (Q, . ) is zero endomorphism, i.e., s(x) = 0 for all x equation Q, then (Q,. ) is an unipotent quasigroup, (Q, . ) equation equation, where equation, (Q, +) is a group, equationAut(Q; +).

(2) If the endomorphism s of a right semimedial quasigroup (Q, . ) is zero endomorphism, i.e., s(x) = 0 for all x equation Q, then (Q, . ) is an unipotent quasigroup, (Q, . ) equation equation, where equation

, (Q, +) is a group, equation Aut(Q, +).

(3) If the endomorphism f of a left E-quasigroup (Q, . ) is zero endomorphism, i.e., f(x) = 0 for all x Q, then up to isomorphism (Q, . ) is a left loop, x.y = x+y, (Q, +) is an Abelian group, α0 = 0.

(4) If the endomorphism e of a right E-quasigroup (Q, . ) is zero endomorphism, i.e., e(x) = 0 for all x equation Q, then up to isomorphism (Q, . ) is a right loop, x.y = x + βy, (Q, +) is an Abelian group, β0 = 0

Proof. (1) We can rewrite equality xx.yz = xy.xz in the form k. yz = xy. xz, where s(x) = k for all x equation Q. If we denote xz by v, then equation and equality k. yz = xy. xz takes the form equation.

Then the last equality has the form A1(y;A2(x; v)) = A3(A4(y; x); v), where A1, A2, A3, A4 are quasigroup operations, namely, equation, u = A4(y; x) = x.y.

From Four quasigroups theorem (Theorem 1.92) it follows that quasigroup (Q, . ) is an isotope of a group (Q, +).

If in the equality k.yz = xy.xz we fix the variable x = b, then we obtain the equality k.yz = by.bz, k.yz = Lby.Lbz. From Theorem 1.97 it follows that (Q, . ) is a right linear quasigroup.

If in k.yz = xy.xz we put x = z, then we obtain k.yx = xy. k. From Lemma 1.99 it follows that (Q, +) is a commutative group.

From Lemma 1.95 we have that there exists a group (Q, +) such that x. y = αx+ ψy+c, where α is a permutation of the set Q, α0 = 0, equationAut(Q, +).

Further we have s(0) = k = 0.0 = c, k = c. Then s(x) = k = x.x = αx + ψx + k. Therefore αx + ψx = 0 for all x equation Q. Then α = Iψ , where x + I(x) = 0 for all x equation Q. Therefore α is an antiautomorphism of the group (Q, +), x.y = Iψ x + ψy + k.

equation is an inner automorphism of (Q, +). It is easy to see that equation for all x equation Q.

Below we will suppose that any left semimedial quasigroup (Q, . ) with zero endomorphism s is an unipotent quasigroup with the form equation, where (Q, +) is a group, ' equation Aut(Q, +).

(2) We can rewrite equality zy.s(x) = zx.yx in the form zy.k = zx.yx, where s(x) = k. If we denote zx by v, then equation and the equality zy.k = zx.yx takes the form equation equation

We rewrite the last equality in the form A1(A2(v; x); y) = A3(v;A4(x; y)), where A1, A2, A3, A4 are quasigroup operations, namely, A1(t; y) = Rk(t.y), t = equation, A3(v; u) = v. u, u = A4(x; y) = x ∗ y.

From Four quasigroups theorem it follows that quasigroup (Q, . ) is an isotope of a group (Q, +).

If in the equality zy.k = zx.yx we fix the variable x = b, then we obtain the equality zy.k = zb.yb, zy.k = Rbz.Rby. From Theorem 1.97 it follows that (Q, . ) is a left linear quasigroup.

If in the equality zy.k = zx.yx we put x = z, then we obtain xy.k = k.yx. Thus from Lemma 1.99 it follows that (Q, +) is a commutative group.

From Lemma 1.95 we have that there exists a group (Q, +) such that x.y = ψx+βy +c, where β is a permutation of the set Q, β = 0, equation Aut(Q, +).

Further we have s(0) = k = 0.0 = c, k = c. Then s(x) = k = x.x = ψx + βx + k. Therefore αx + βx = 0 for all x equation Q. Then β = Iψ, where Ix + x = 0 for all x equation Q.

Therefore β = Iψ equation Aut(Q, +), x.y = ψx−ψy + k.

We have equation. It is easy to see that equation for all x equation Q.

Below we will suppose that any right semimedial quasigroup (Q, ) with zero endomorphism s is an unipotent quasigroup with the form x.y = ψx−ψy, where (Q, +) is a group, equation Aut(Q; +).

(3) We can rewrite the equality x. yz = f(x)y.xz in the form x.yz = ky. xz = y. xz, x.(z ∗ y) = xz ∗ y, where f(x) = k for all x equation Q.

Then A1(x;A2(z; y)) = A3(A4(x; z); y), where A1, A2, A3, A4 are quasigroup operations, namely, A1(x; t) = x . t, t = A2(z; y) = z ∗ y, A3(u; y) = u ∗ y, u = A4(x; z) = x . z. From Four quasigroups theorem it follows that quasigroup (Q, . ) is a group isotope.

If in the equality x.yz = y.xz we fix variable z, i.e., if we take z = a, then we have x.Ray = y.Rax. From Corollary 1.94 it follows that the group (Q, +) is commutative.

If in the equality x.yz = y.xz we fix variable x, i.e., if we take x = a, then we have a yz = y.az, a.(yz) = y.Laz. The application of Theorem 1.97 to the last equality gives us that (Q, ) is a right linear quasigroup, i.e., x.y = αx + ψy + c.

Then f(x).x = k.x = αk + ψx + c =x. By x = 0 we have αk + ψ0 + c = 0, αk = −c. Therefore, k.x = x = x for all x equation Q. Then ψ = ε x.y = αx + y + c = Lcαx + y for all x; y equation Q. In other words, x.y = αx + y for all x; yequation Q.

Further let a + α0 = 0. Then equation equation

(4) Case (4) is a "mirror" case of Case (3), but we give the direct proof. We can rewrite equality zy.x = zx.ye(x) in the form zy.x = zx.yk = zx. y, (y ∗ z).x = y ∗ zx, where e(x) = k.

Then A1(A2(y; z); x) = A3(y;A4(z; x)), where A1, A2, A3, A4 are quasigroup operations, namely, A1(t; x) = t.x, t = A2(y; z) = y ∗z, A3(y; v) = y ∗ v, v = A4(z; x) = z.x.

From Four quasigroups theorem it follows that quasigroup (Q, . ) is an isotope of a group (Q, +).

If in the equality zy.x = zx.y we fix variable z, i.e., if we take z = a, then we have Lay.x = Lax.y. From Corollary 1.94 it follows that the group (Q, +) is commutative.

If in the equality zy.x = zx.y we fix variable x, i.e., if we take x = a, then we have zy.a = za.y, zy.a = Raz.y. The application of Theorem 1.97 to the last equality gives us that (Q, ) is a left linear quasigroup, i.e., x.y = ψx + βy + c.

Then x.e(x) = x.k = ψx + βk + c = x. By x = 0 we have ψ0 + ψk + c = 0, βk = −c. Therefore x.k = x = ψx for all x equation Q. Then β = ", x.y = x + βy + c = x + Rcβy for all x; y equation Q. In other words, x.y = x + βy for all x; y equation Q.

Further let a + β0 = 0. Then equation equation

In proof of the following lemma we use ideas from [15].

Lemma 2.15. (1) If the endomorphism s of a left semimedial quasigroup (Q, . ) is a permu- tation of the set Q, then quasigroup equation of the form equation is a left distributive quasigroup and s equation Autequation.

(2) If the endomorphism s of a right semimedial quasigroup (Q, . ) is a permutation of the set Q, then quasigroup equation of the form equation is a right distributive quasigroup and s equation Aut equation.

(3) If the endomorphism f of a left E-quasigroup (Q, ) is a permutation of the set Q, then quasigroup equation of the form x ο y = f(x). y is a left distributive quasigroup and f equation Autequation.

(4) If the endomorphism e of a right E-quasigroup (Q, ) is a permutation of the set Q, then quasigroup equation of the form x ο y = x. e(y) is a right distributive quasigroup and e equation Autequation.

Proof. (1) We prove that equation is left distributive. It is clear that s−1 equation Aut(Q, . ). We have

equation

equation

Prove that s equation Autequation. We have equation. See also [72].

(2) We prove that equation is right distributive. It is clear that equation. We have

equation

equation

Prove that s equation Autequation We have equation

(3) If the endomorphism f is a permutation of the set Q, then f; equation. We have

equation

equation

Prove that f equation Aut equation. We have f(x ο y) = f(f(x).y) = f2(x). f(y) = f(x) ο f(y). (4) If the endomorphism e is a permutation of the set Q, then e; e−1 equation Aut(Q, . ). We have

equation

Prove that e equation Aut equation. We have e(x ο y) = e(x.e(y)) = e(x).e2(y) = e(x) ο e(y).

Remark 2.16. By the proof of Lemma 2.15 it is possible to use Lemma 2.3 and parastrophe invariant arguments.

Theorem 2.17. (1) Every left SM-quasigroup (Q, . ) has the following structure:

equation

where (A; ο) is a quasigroup with a unique idempotent element and there exists a number m such that equation is an isotope of a left distributive quasigroup (B; ?), x.y = s(x? y) for all x; y 2 B, s equation Aut(B; ), s equation Aut(B; ?).

(2) Every right SM-quasigroup (Q, . ) has the following structure:

equation

where (A; ο) is a quasigroup with a unique idempotent element and there exists an ordinal number m such that jsm(A; ο)j = 1; (B, . ) is an isotope of a right distributive quasigroup (B, ?), x.y = s(x ? y) for all x; y equation B, s equation Aut(B, . ), s equation Aut(B;?).

(3) Every left E-quasigroup (Q, . ) has the following structure:

equation

where (A; ο) is a quasigroup with a unique idempotent element and there exists a number m such that equation = 1; (B, . ) is an isotope of a left distributive quasigroup (B; ?), x.y = f−1(x) ? y for all x; y equation B, f equation Aut(B, . ), f equation Aut(B, ?).

(4) Every right E-quasigroup (Q, . ) has the following structure:

equation

where (A; ο) is a quasigroup with a unique idempotent element and there exists a number m such that jem(A; ο)j = 1; (B, . ) is an isotope of a right distributive quasigroup (B; ?), x.y = x ? e−1(y) for all x; y equation B, e equation Aut(B, . ), e equation Aut(B; ?).

Proof. The proof is similar to the proof of Theorem 2.7. It is possible also to use parastrophe invariance ideas.

Corollary 2.18. If (Q, . ) is a left SM-quasigroup, then sm(Q, . ) equation (Q, . ); if (Q, . ) is a right SM-quasigroup, then sm(Q, . )) P (Q, . ); if (Q, . ) is a left E-quasigroup, then fm(Q, . ) equation(Q, . ); if (Q, . ) is a right E-quasigroup, then em(Q, . ) equation (Q, . ).

Corollary 2.19. If (Q, . ) is a left SM-quasigroup with an idempotent element, then equiv- alence class equation of the normal congruence Ker sj containing an idempotent element a equation Q forms an unipotent quasigroup (equation, . ) isotopic to a group with isotopy of the form equation, where equation Aut(equation, . ) for all suitable values of j.

If (Q, . ) is a right SM-quasigroup with an idempotent element, then equivalence class equation of the normal congruence Ker sj containing an idempotent element a equation Q forms an unipotent quasigroup (equation, . ) isotopic to a group with isotopy of the form equation, where ' equation Aut(equation, . ) for all suitable values of j.

If (Q, . ) is a left E-quasigroup with an idempotent element, then equivalence class equation of the normal congruence Ker fj containing an idempotent element a equation Q forms a left loop isotopic to an Abelian group with isotopy of the form equation, for all suitable values of j.

If (Q, . ) is a right E-quasigroup with an idempotent element, then equivalence class equation of the normal congruence Ker ej containing an idempotent element a equation Q forms a right loop isotopic to an Abelian group with isotopy of the form equation for all suitable values of j.

Proof. Mainly the proof repeats the proof of Corollary 2.10. It is possible to use Theorem 2.14.

CML as an SM-quasigroup

In this subsection we give information (mainly well known) about commutative Moufang loops (CML) which is possible to obtain from the fact that a loop (Q, . ) is left semimedial if and only if it is a commutative Moufang loop. Novelty of information from this subsection is in the fact that some well-known theorems about CML are obtained quit easy using quasigroup approach.

Lemma 2.20. (1) A left F-loop is a group. (2) A right F-loop is a group. (3) A loop (Q, . ) is left semimedial if and only if it is a commutative Moufang loop. (4) A loop (Q, . ) is right semimedial if and only if it is a commutative Moufang loop. (5) A left E-loop (Q; ) is a commutative group. (6) A right E-loop (Q, . ) is a commutative group.

Proof. (1) From x.yz = xy.e(x)z we have x.yz = x.yz.

(2) From xy.z = xf(z).yz we have xy.z = x.yz.

(3) We use the proof from [12, p. 99]. Let (Q, . ) be a left semimedial loop. If y = 1, then we have x2.z = x.xz. If z = 1, then x2y = xy.x. Then x.xy = xy.x. If we denote xy by y, then we obtain that xy = yx, i.e., the loop (Q, . ) is commutative.

It is clear that a commutative Moufang loop is left semimedial.

(4) For the proof of Case (3) it is possible to use \mirror" principles. We give the direct proof. Let (Q, . ) be a right semimedial loop, i.e., zy.x2 = zx.yx for all x; y; z equation Q. If y = 1, the we have z.x2 = zx.x.

If z = 1, then yx2 = x.yx. Then zx.x = x.zx. If we denote zx by z, then we obtain that zx = xz, i.e., the loop (Q, . ) is commutative. Moreover, we have x2.yz = zy.x2, xy.zx = xz.yx.

It is clear that a commutative Moufang loop is right semimedial.

(5) From x.yz = f(x)y.xz we have x.yz = y.xz. From the last identity by z = 1 we obtain x.y = y.x. Therefore we can rewrite identity x.yz = y. xz in the form yz.x = y.zx.

Case (6) is proved in the similar way to Case (5).

Commutative Moufang loop in which any element has the order 3 is called 3-CML.

Remark 2.21. Center C(Q, +) of a CML (Q, +) is a normal Abelian subgroup of (Q; +) and it coincides with the left nucleus of (Q, +) [12, 24].

Lemma 2.22. In a commutative Moufang loop the map δ : x → 3x is central endomorphism [12, 24].

Proof. In a CML (Q; +) we have n(x + y) = nx + ny for any natural number n since by Moufang theorem [12,24,76] CML is diassociative (any two elements generate an associative subgroup). Therefore the map ο is an endomorphism. See [65] for many details on commutative diassociative loops.

The proof of centrality of the endomorphism ο is standard [83,12,24,56] and we omit it.

A quasigroup (Q, . ) with identities xy = yx, x.xy = y, x.yz = xy. xz is called a distributive Steiner quasigroup [12,15].

Theorem 2.23. (1) Every commutative Moufang loop (Q; +) has the following structure:

equation

where (A; equation) is an Abelian group and there exists a number m such that equation; (B; +) is an isotope of a distributive quasigroup (B; ?), x + y = s(x ? y) for all x; y equation B, s equation Aut(B; +), s equation Aut(B; ?).

(2) C(Q; +) ≅ equation × C(B; +).

(3) (Q; +)=C(Q; +) ≅ (B; +)=C(B; +) ≅ (D; +) is 3-CML in which the endomorphism s is permutation I such that Ix = −x.

(4) Quasigroup (D; ?), x ? y = −x−y, x; y equation (D; +), is a distributive Steiner quasigroup.

Proof. (1) The existence of decomposition of (Q; +) into two factors follows from Theorem 2.17.

From Corollary 2.19 it follows that any equivalence class equation = Hj of the normal congruence Ker sj containing an idempotent element 0 equation Q is an unipotent loop (Hj ; . ) isotopic to

Since (Hj,.) is a commutative loop, we have that equation equation Thus x .y = x + y for all x, y (Hj,.).

We notice in a commutative Moufang loop (Q, +) the map si takes the form 2i, i.e., si.(x) = 2i(x). Then in the loop (A;equation) any nonzero element has the order 2i or infinite order.

If an element x of the loop (A;equation) has a finite order, then x ∈ C(A;equation), where C(A;equation) is a center of (A;equation) since G.C.D:(2i, 3) = 1.

If an element x of the loop (A;equation) has infinite order, then by Lemma 2.22 3x ∈ C(A;equation), ? x ? & ≅ ? 3x ?.

Therefore (A,equation) ≅ 3(A,equation) ⊆ C(A,equation), (A,equation) is an Abelian group.

From Theorem 2.17(1), (2) it follows that (B, +) is an isotope of left and right distributive quasigroup. Therefore, (B, +) is an isotope of distributive quasigroup.

(2) From Lemma 1.74 it follows that C(Q, +) ≅ C(A,equation)*C(B, +). Therefore C(Q, +) ≅ (A,equation) * C(B, +) since C(A,equation) = (A,equation).

(3) The fact that (Q, +)=C(Q, +) is 3-CML is well known and it follows from Lemma 2.22. Isomorphism (Q, +)=C(Q, +) ≅ ((A,equation)*(B, +))=((A,equation)*C(B, +)) follows from Cases (1) and (2).

Isomorphism

equation

follows from the Second Isomorphism Theorem (see [27, p. 51], for group case see [47]) and the fact that equation

(4) It is clear that in 3-CML (D, +) the map s takes the form s(x) = 2x = -x = Ix. Moreover, I-1 = I. It is easy to see that the quasigroup (D; *) is a distributive Steiner quasigroup.

Corollary 2.24. If in CML (Q, +) the endomorphism s has finite order m, then (i) any nonzero element of the group (A,equation) has the order 2i, 1 ≤ i ≤ m; (ii) Aut(Q, +) ≅ Aut(A,equation) * Aut(B, +).

Proof. (i). It is easy to see.

(ii). Let (Q, +) be a commutative Moufang loop, α ∈ Aut(Q, +). Then the order of an element x coincides with the order of element α(x). Indeed, if nx = 0, then n(αx) = α(nx)=0.

The loops (A,equation) and (B, +) have elements of different orders. Indeed, the orders of elements of the loop (A,equation) are powers of the number 2 and orders of the elements of the loop (B, +) are some odd numbers or, possibly, ∞

Therefore loops (A,equation) and (B, +) are invariant relative to any automorphism of the loop (Q, +). Then Aut(Q, +) ≅ Aut(A,equation) * Aut(B, +).

The structure

Theorems 2.7 and 2.17 give us a possibility to receive some information on left and right F-, SM-, E-quasigroups and some combinations of these classes.

Simple left and right F-, E-, and SM-quasigroups

Simple quasigroups of some classes of finite left distributive quasigroups are described in [39]. The structure and properties of F-quasigroups are described in [55, 56]. We give Jezek-Kepka Theorem [46] in the following form [101, 103].

Theorem 3.1. If a medial quasigroup (Q,.) of the form x.y = αx+βy+a over an Abelian group (Q, +) is simple, then

(1) the group (Q, +) is the additive group of a finite Galois field GF(pk);

(2) the group ? α,β ? is the multiplicative group of the field GF(pk) in the case k > 1, the group ? α,β ? is any subgroup of the group Aut(Zp, +) in the case k = 1;

(3) the quasigroup (Q,.) in the case equation can be quasigroup from one of the following disjoint quasigroup classes:

(a) α + β = ε, a = 0; in this case the quasigroup (Q,.) is an idempotent quasigroup;

(b) α + β = ε and a ≠ 0; in this case the quasigroup (Q,.) does not have any idempotent element, the quasigroup (Q,.) is isomorphic to the quasigroup (Q,*) with the form x*y= αx + βy + 1 over the same Abelian group (Q, +);

(c) α + β ≠ ε; in this case the quasigroup (Q,.) has exactly one idempotent element, the quasigroup (Q,.) is isomorphic to the quasigroup (Q, o) of the form x0y = αx + βy over the group (Q, +).

Theorem 3.2. If a simple distributive quasigroup (Q,o) is isotopic to finitely generated commutative Moufang loop (Q, +), then (Q, o) is a nite medial distributive quasigroup [94, 95, 96].

Proof. It is known [24] that any finitely generated CML (Q, +) has a nonidentity center C(Q, +) (for short C).

We check that center of CML (Q, +) is invariant (is a characteristic subloop) relative to any automorphism of the loop (Q, +) and the quasigroup (Q, o).

Indeed, if equation(see Remark 2.21), then we have equation equation thus equation .For any distributive quasigroup (Q, o) of the form equation we have

equation

where I is the group of inner permutations of commutative Moufang loop (Q, +),

equation

Therefore any automorphism of (Q,o) has the form L+a α, where α ∈ Aut(Q, +) [97].

The center C defines normal congruence θ of the loop (Q, +) in the following way: xθy ⇔ x + C = y + C. We give a little part of this standard proof: (x + a) + C = (y + a) + C ⇔ (x + C) + a = (y + C) + a ⇔ x + C = y + C. In fact C is coset class of θ containing zero element of (Q, +).

The congruence θ is admissible relative to any permutation of the form L+a α, where α ∈ Aut(Q, +), since θ is central congruence. Therefore, θ is congruence in the quasigroup (Q, o).

Since (Q, o) is simple quasigroup and θ cannot be diagonal congruence, then θ = Q * Q, C(Q, +) = (Q, +), (Q, o) is medial. From Theorem 3.1 it follows that (Q,o) is nite.

We notice it is possible to prove Theorem 3.2 using Theorem 1.63 [96].

Lemma 3.3. If simple quasigroup (Q,.) is isotope either of the form (f,ε,ε), or of the form (ε,e,ε), or of the form (ε,ε,s) of a distributive quasigroup (Q,o), where f, e, s ∈ Aut(Q, o) and (Q, o) is isotopic to finitely generated commutative Moufang loop (Q,.+), then (Q,.) is finite medial quasigroup.

Proof. Since (Q, +) is finitely generated, then equation [24]. From the proof of Theorem 3.2 it follows that C(Q, +) is invariant relative to any automorphism of (Q,o).

Therefore necessary condition of simplicity of (Q, ) is the fact that C(Q, +) = (Q, +). Then (Q, o) is medial.

Prove that (Q,.) is medial, if x.y = fx o y. We have xy.uv = f(fx o y) o (fu o v) =(f2x o fu) o (fy o v) = (xu).(yv) [78].

Prove that (Q,.) is medial, if x.y = x o ey. We have xy.uv = (x o ey) o (eu o e2v) =(x o eu) o (ey o e2v) = (xu).(yv) [78].

Prove that (Q,.) is medial, if x.y = s(x o y). We have xy.uv = (s2x o s2y) o (s2u o s2v) =(s2x o s2u)o (s2y o s2v) = (xu).(yv) [78].

We can obtain some information on simple left and right F-, E-, and SM-quasigroups.

Theorem 3.4. (1) Left F-quasigroup (Q,.) is simple if and only if it lies in one from the following quasigroup classes:

(i) (Q,.) is a right loop of the form x.y = x +ψy, where ψ ∈ Aut(Q, +) and the group (Q, +) is ψ-simple;

(ii) (Q,.) has the form x.y = x o ψy, where ψ∈ Aut(Q, o) and (Q, o) is ψ-simple left distributive quasigroup.

(2) Right F-quasigroup (Q,.) is simple if and only if it lies in one from the following quasigroup classes:

(i) (Q,.) is a left loop of the form equation where equation and the group (Q, +) is equation-simple;

(ii) (Q,.) has the form x.y = equationx o y, where equation∈ Aut(Q; o) and (Q; o) is equation -simple left distributive quasigroup.

(3) Left SM-quasigroup (Q,.) is simple if and only if it lies in one from the following quasigroup classes:

(Q,.) is a unipotent quasigroup of the form equation(Q, +) is a group, equation ∈ Aut(Q, +) and the group (Q, +) is equation equation -simple;

(ii) (Q,. ) has the form x .y = equation (x 0 y), where equation ∈ Aut(Q, 0) and (Q, 0) is equation-simple left distributive quasigroup.

(4) Right SM-quasigroup (Q,.) is simple if and only if it lies in one from the following quasigroup classes:

(i) (Q,.) is a unipotent quasigroup of the form x o y = equation x o equation y, (Q, +) is a group, equation∈ Aut(Q, +) and the group (Q,+) is equation-simple;

(ii) (Q,.) has the form x.y = equation (x o y), where equation ∈ Aut(Q, o) and (Q, o) is equation-simple right distributive quasigroup.

(5) Left E-quasigroup (Q,.) is simple if and only if it lies in one from the following quasigroup classes:

(i) (Q,.) is a left loop of the form x.y = αx + y, α0 = 0, and (Q, +) is α-simple Abelian group;

(ii) (Q,.) has the form x.y = equation x o y, where equation ∈ Aut(Q, o) and (Q, o) is equation-simple left distributive quasigroup.

(6) Right E-quasigroup (Q,.) is simple if and only if it lies in one from the following quasigroup classes:

(i) (Q,.) is a right loop of the form x.y = x+βy, β0 = 0, and (Q; +) is β-simple Abelian group;

(ii) (Q,.) has the form x  y = x o yψ, where ψ∈ Aut(Q, o) and (Q, o) is ψ-simple right distributive quasigroup.

Proof. (1) Suppose that (Q,.) is simple left F-quasigroup. From Theorem 2.7 it follows that (Q,.) can be a quasigroup with a unique idempotent element or an isotope of a left distributive quasigroup.

By Theorem 1.51 the endomorphism e de nes the corresponding normal congruence Ker e. Since (Q,.) is simple, then this congruence is the diagonal equation or the universal congruence Q * Q.

From Theorem 2.7 it follows that in simple left F-quasigroup the map e is zero endomorphism or a permutation.

Structure of left F-quasigroups in the case when e is zero endomorphism follows from Lemma 2.2.

Structure of left F-quasigroups in the case when e is an automorphism follows from Lemma 2.3. Additional properties of quasigroup (Q, o) follow from Lemma 1.62.

Conversely, using Corollary 1.70 we can say that left F-quasigroups from these quasigroup classes are simple.

Cases (2){(6) are proved in a similar way.

Remark 3.5. Left F-quasigroup (Z,.), where x.y = -x + y, (Z, +) is the infinite cyclic group, (Example 2.6) is not simple. Indeed, in this quasigroup the endomorphism e is not a permutation (a bijection) of the set Z or a zero endomorphism.

We can also apply Theorem 3.4(3), since (Z,.) is a left SM-quasigroup, and so on.

F-quasigroups

Simple F-quasigroups isotopic to groups (FG-quasigroups) are described in [56]. The authors prove that any simple FG-quasigroup is a simple group or a simple medial quasigroups. We notice that simple medial quasigroups are described in [46]. See also [101, 103]. Conditions when a group isotope is a left (right) F-quasigroup are there in [66, 113].

The following examples demonstrate that in an F-, E-, SM-quasigroup the order of map e does not coincide with the order of map f, i.e., there exists some independence of the orders of maps e, f, and s.

Example 3.6. By (Z3; +) we denote the cyclic group of order 3 and we take Z3 = {0, 1, 2}. Groupoid (Z3;), where x.y = x-y, is a medial E-, F-, SM-quasigroup and e(Z3) = s(Z3) = {0}, f(Z3) = Z3.

Example 3.7. By (Z6, +) we denote the cyclic group of order 6 and we take Z6 = {0; 1; 2; 3; 4; 5}. Groupoid (Z6,.), where x  y = x - y, is a medial E-, F-, SM-quasigroup and e.(Z6) = s.(Z6) = {0}, f(Z6) = {0, 2, 4}.

The following lemmas give connections between the maps e and f in F-quasigroups.

Lemma 3.8. (1) Endomorphism e of an F-quasigroup (Q,.) is zero endomorphism, i.e., e(x) = 0 for all x ∈ Q if and only if x.y = x+ y, (Q, +) is a group, ψ∈ Aut(Q, +), (Q,.) contains unique idempotent element 0, x + fy = fy + x for all x, y ∈ Q.

(2) Endomorphism f of an F-quasigroup (Q; ) is zero endomorphism, i.e., f(x) = 0 for all x ∈ Q if and only if x.y = equationx + y, (Q, +) is a group, equation∈ Aut(Q, +), (Q,.) contains unique idempotent element 0, x + ey = ey + x for all x, y ∈ Q.

Proof. (1) From Lemma 2.2(1) it follows that (Q,.) is a right loop, isotope of a group (Q, +) of the form x .y = x + ψy, where ψ∈ Aut(Q,+).

If a.a = a, then a + ψa = a, a = 0, a = 0.

If we rewrite right F-quasigroup equality in terms of the operation +, then we obtain equation If we take y = 0 in the last equality, then equation Therefore equation equation

Conversely, from x.y = x + y we have equation for all x ∈ Q.

(2) This case is proved in a similar way to Case (1).

Lemma 3.9. (1) If endomorphism e of an F-quasigroup (Q,.) is zero endomorphism, i.e., e(x) = 0 for all x ∈ Q, then

(i) f(x) = x - ψx, f ∈ End(Q; +);

(ii) f(Q, +) ⊆ C(Q, +);

(iii)equation, where (H, +) is equivalence class of the congruence Ker f containing identity element of (Q, +)

(iv) f(Q,.) is a medial F-quasigroup; (H,.) = (H, +) is a group; (equation ; ), where equation is equiva- lence class of the normal congruence Ker fj containing an idempotent element a ∈ Q, i ≥ 1, is an Abelian group.

(2) If endomorphism f of an F-quasigroup (Q,.) is zero endomorphism, i.e., f(x) = 0 for all x ∈ Q, then

(i)equation

(ii)e(Q, +) ⊆ C(Q, +);

(iii) equation where (H, +) is equivalence class of the congruence Ker e containing identity element of (Q, +);

(iv)e(Q,.) is a medial F-quasigroup; (H,.) = (H,. +) is a group; (equation;), where equation is equiva- lence class of the normal congruence Ker ej containing an idempotent element a 2 Q, i ≥ 1, is an Abelian group.

Proof. (1) (i) From Lemma 3.8(1) we have equation We can rewrite equality f(x. y) = f(x) . f(y) in the form equationthen we have f(0) = 0. If x = 0, then equation.Therefore

equation (3.1)

(ii) If we apply to equality (3.1) the equality equationthen we obtain equation equation equationi.e., equation

(iii) From de nitions and Case (ii) it follows that equationThe last follows from de nition of (H, +).

(iv) f(Q,.) is a medial F-quasigroup since from Case (ii) it follows that f(Q,+) is an Abelian group. Quasigroup (H,. ) is a group since in this quasigroup the maps e and f are zero endomorphisms and we can use Case (i).

equation is an Abelian group since in this quasigroup the maps e and f are zero endomorphisms and fi(Q,.) is a medial quasigroup for any suitable value of the index i. Moreover, it is well known that a medial quasigroup any its subquasigroup is normal [60]. Then equation

(2) This case is proved in a similar way to Case (1).

Corollary 3.10. Both endomorphisms e and f of an F-quasigroup (Q,.) are zero endomor- phisms if and only if (Q,.) is a group.

Proof. By Lemma 3.8(1) x.y = x+ ψy. By Lemma 3.9(1)(i), f(x) = x-ψx. Since f(x) = 0 for all x ∈Q, further we have ψ = ε.

Conversely, it is clear that in any group e(x) = f(x) = 0 for all x ∈ Q.

Example 3.11. By (Z4, +) we denote the cyclic group of order 4 and we take Z4 = {0; 1; 2; 3}. Groupoid (Z4,), where x.y = x + 3y, is a medial E-, F-, SM-quasigroup, e(Z4) = s(Z4) = {0} and f(Z4) = {0;2} = H.

Corollary 3.12. (1) If in F-quasigroup (Q,.) endomorphism e is zero endomorphism and the group (Q, +) has identity center, then (Q,.) = (Q, +).

(2) If in F-quasigroup (Q,.) endomorphism f is zero endomorphism and the group (Q, +) has identity center, then (Q,.) = (Q, +).

Proof. The proof follows from Lemma 3.9(ii), (iii).

Corollary 3.13. (1) If endomorphism e of an F-quasigroup (Q,.) is zero endomorphism, i.e., e(x) = 0 for all x ∈ Q, endomorphism f is a permutation of the set Q, then x.y = x+ y, (Q, +) is an Abelian group, 2 Aut(Q, +) and (Q, o), xoy = fx+ψ y, is a medial distributive quasigroup.

(2) If endomorphism f of an F-quasigroup (Q,.) is zero endomorphism, i.e., f(x) = 0 for all x ∈ Q, endomorphism e is a permutation of the set Q, then x.y = equationx + y, (Q; +) is an Abelian group, equation∈ Aut(Q, +) and (Q, o), x o y = equationx + ey, is a medial distributive quasigroup.

Proof. The proof follows from Lemma 3.9. It is a quasigroup folklore that idempotent medial quasigroup is distributive [91, 92].

Remark 3.14. It is easy to see that condition "(D,) is a medial F-quasigroup of the form x. y = x + ψy such that (D, o), x o y = fx + ψ y, is a medial distributive quasigroup" in Corollary 3.13 is equivalent to the condition that the automorphism ψ of the group (D, +) is complete (Definition 1.32).

Lemma 3.15. (1) If endomorphism e of an F-quasigroup (Q,⋅) is a permutation of the set Q, i.e., e is an automorphism of (Q,⋅), then (Q, o), x o y = x⋅e(y), is a left distributive quasigroup which satis es the equality (x o y) o z = (x o fz) o (y o e-1z), for all x, y, z ∈ Q. (2) If endomorphism f of an F-quasigroup (Q,⋅) is a permutation of the set Q, i.e., f is an automorphism of (Q,⋅), then (Q, o), x o y = f(x)⋅ y, is a right distributive quasigroup which satis es the equality x o (y o z) = (f-1x o y) o (ex o z), for all x, y, z ∈ Q.

Proof. (1) The fact that (Q, o), x o y = xe(y), is a left distributive quasigroup follows from Lemma 2.3. If we rewrite right F-quasigroup equality in terms of the operation o, then (xoe-1y)oe-1z = (xoe-1fz)o(e-1y oe-2z). If we replace e-1y by y, e-1z by z and take into consideration that e-1f = fe-1, then we obtain the equality (xo y) o z = (xo fz) o (y o e-1z).

(2) The proof is similar to Case (1).

Corollary 3.16. (1) If endomorphism e of an F-quasigroup (Q,⋅) is identity permutation of the set Q, then (Q,⋅) is a distributive quasigroup.

(2) If endomorphism f of an F-quasigroup (Q,⋅) is identity permutation of the set Q, then (Q,⋅) is a distributive quasigroup.

Proof. (1) If fxx = x, then fxoe-1x = x. Further proof follows from Lemma 3.15. Indeed from fx o e-1x = x it follows fx o x = x, fx = x, since (Q, o) is idempotent quasigroup. Then f = ε.

(2) The proof is similar to Case (1).

The following proof belongs to the OTTER 3.3 [73]. We also have used much of Phillipsφ article [85]. Here we give the adopted (humanized) form of this proof.

Theorem 3.17. If in a left distributive quasigroup (Q, o) the equality

(x o y) o z = (x o fz) o (y o ez) (3.2)

is ful lled for all x, y, z ∈ Q, where f, e are the maps of Q, then the following equality is ful lled in (Q, o): (x o y) o fz = (x o fz) o (y o fz).

Proof. If we pass in equality (3.2) to operation /, then we obtain

equation

from equality (3.2) by x = y we obtain xoz = (xofz)o(xoez) and using left distributivity we have x o z = x o (fz o e(z)),

z = fz o e(z), e(z) = fz \z (3.4)

If we change in equality (3.3) the expression e(z) using equality (3.4), then we obtain

equation

We make the following replacements in (3.5): x → x/z, y → z, z → y. Then we obtain (x o y) o z → ((x=z) o z) o y = x o y and the following equality is fulfilled:

equation

Using the operation / we can rewrite left distributive identity in the following form:

equation

If we change in identity (3.7) (y oz) by y, then variable y passes in y/z. Indeed, if y oz = t, then y = t/z. Therefore, we have

equation

from equality (3.2) using left distributivity to the right-hand side of this equality we obtain (x o y) o z = ((x o fz) o y) o ((x o fz) o ez). After applying of the operation / to the last equality we obtain

equation

After substitution of (3.4) in (3.9) we obtain

equation

Now we show the most unexpected OTTER's step. We apply the left-hand side of equality (3.6) to the left-hand side equality (3.10). In this case expression ((x o y) o z) from (3.10) plays the role of (x o y), (x o f(z)) the role of z, and (fz \ z) the role of (f(y)\y).

Therefore we obtain

equation

After application to the left-hand side of equality (3.11) equality (3.8) we have

equation

If we change in equality (3.12) (y/fz) by y, then variable y passes in y o fz. Therefore (x o y) o fz = (x o fz) o (y o fz).

Corollary 3.18. If in a left distributive quasigroup (Q, o) the equality

(x o y) o z = (x o fz) o (y o ez)

is ful lled for all x, y, z ∈ Q, where e is a map, f is a permutation of the set Q, then (Q, o) is a distributive quasigroup.

Proof. The proof follows from Theorem 3.17.

Theorem 3.19. If in F-quasigroup (Q,⋅) endomorphisms e and f are permutations of the set Q, then (Q,⋅) is isotope of the form x⋅y = x o e-1y of a distributive quasigroup (Q, o).

Proof. Quasigroup (Q, o) of the form x o y = x e(y) is a left distributive quasigroup (Lemma 2.3) in which the equality (x o y) o z = (x o fz) o (y o e-1z), is true (Lemma 3.15). By Corollary 3.18 (Q, o) is distributive.

Theorem 3.20. An F-quasigroup (Q,⋅) is simple if and only if (Q,⋅) lies in one from the following quasigroup classes:

(i) (Q,⋅) is a simple group in the case when the maps e and f are zero endomorphisms,

(ii) (Q,⋅) has the form x⋅y = x + ψ y, where (Q, +) is a ψ-simple Abelian group, ∈ Aut(Q, +), in the case when the map e is a zero endomorphism and the map f is a permutation, in this case e = -ψ , fx + ψx = x for all x ∈ Q,

(iii) (Q,⋅) has the form x⋅y = φx + y, where (Q, +) is a φ-simple Abelian group, φ ∈ Aut(Q, +), in the case when the map f is a zero endomorphism and the map e is a permutation, in this case f = -φ, φx + ex = x for all x ∈ Q,

(iv) (Q,⋅) has the form x⋅y = x o ψy, where (Q, o) is a ψ-simple distributive quasigroup ∈ Aut(Q, o), in the case when the maps e and f are permutations, in this case e = ψ-1, fx o ψx = x for all x ∈ Q.

Proof. (⇒) (i) It is clear that in this case left and right F-quasigroup equalities are transformed in the identity of associativity.

(ii) from Lemma 3.9 (iii) and the fact that the map f is a permutation of the set Q it follows that (Q, +) is an Abelian group.

(iii) This case is similar to Case (ii).

(iv) By Belousov result [15] (see Lemma 2.3 of this paper) if the endomorphism e of a left F-quasigroup (Q,⋅) is a permutation of the set Q, then quasigroup (Q,⋅) has the form x⋅y = xoψy, where (Q, o) is a left distributive quasigroup and ∈ Aut(Q, o), ∈ Aut(Q,⋅). The right distributivity of (Q, o) follows from Theorem 3.19.

(⇐) Using Corollary 1.70 we can say that F-quasigroups from these quasigroup classes are simple.

Remark 3.21. There exists a possibility to formulate Theorem 3.20(iv) in the following form.

(iv)* (Q,⋅) has the form x⋅y = φxo y, where (Q, o) is a φ-simple distributive quasigroup, in the case when the maps e and f are permutations, in this case f = φ-1, φx o ex = x for all x ∈ Q.

Corollary 3.22. Finite simple F-quasigroup (Q,⋅) is a simple group or a simple medial quasigroup.

Proof. Theorem 3.20(i) demonstrates that simple F-quasigroup can be a simple group. Taking into consideration Toyoda Theorem (Theorem 1.30) we see that Theorem 3.20(ii),

(iii) provide that simple F-quasigroups can be simple medial quasigroups.

We will prove that in Theorem 3.20(iv) we also obtain medial quasigroups.

The quasigroup (Q,⋅) is isotopic to distributive quasigroup (Q, o), quasigroup (Q, o) is isotopic to CML (Q, +). Therefore (Q,⋅) is isotopic to the (Q, +) and we can apply Lemma 3.3.

Taking into consideration Lemma 1.27 we can say that some properties of nite simple medial F-quasigroups are described in Theorem 3.1.

Using the results obtained in this section we can add information on the structure of F-quasigroups [56].

Theorem 3.23. Any nite F-quasigroup (Q,⋅) has the following structure:

equation

where (A, o) is a quasigroup with a unique idempotent element, (B,⋅) is isotope of a left distributive quasigroup (B,?), xy = x? y, ∈ Aut(B, ), ∈ Aut(B, ?). In the quasigroups (A, o) and (B,⋅) there exist the following chains:

A ⊃ e(A) ⊃⋅⋅⋅⊃ em-1(A) ⊃ em(A) = 0, B ⊃ f(B) ⊃⋅⋅⋅ ⊃ fr(B) = fr+1(B)

where

(1) Let Di be an equivalence class of the normal congruence Ker ei containing an idempo- tent element a ∈ A, i > 0. Then

(a) (Di, o) is linear right loop of the form x o y = x + y, where ∈ Aut(Di, +),

(b) Ker(fj(Di,o)) is a group,

(c) if j > 1, then Ker(fj j(Di,o)) is an Abelian group,

(d) if f is a permutation of fl(Di, o), then fl(Di, o) is a medial right loop of the form x o y = x + y, where is a complete automorphism of the group fl(Di, +),

(e) (Di, o) = (Ei, +)  fl(Di, o), where (Ei, +) is a linear right loop, an extension of an Abelian group by Abelian groups and by a group.

(2) Let Hj be an equivalence class of the normal congruence Ker fj containing an idempo- tent element b ∈ B, j > 0. Then

(a) (H0, ) is a linear left loop of the form x⋅y = φx + y,

(b) f(B, ) is isotope of a distributive quasigroup f(B, ?) of the form x⋅y = x ? e-1y,

(c) if 0 < j < r, then (Hj , ) is medial left loop of the form x⋅y = φx + y, where (Hj , +) is an Abelian group, φ ∈ Aut(Hj , +) and (Hj , ?), x ? y = φx + ey, is a medial distributive quasigroup,

(d) (B, ) = (G, +)  fr(B, ), where (G, +) has a unique idempotent element, is an extension of an Abelian group by Abelian groups and by a linear left loop (H0,⋅), fr(B,⋅) is a distributive quasigroup.

Proof. from Theorem 2.7(1) it follows that F-quasigroup (Q,⋅) is isomorphic to the direct product of quasigroups (A, o) and (B,⋅).

In F-quasigroup (A, o) the chain

A ⊃ e(A) ⊃ e∈(A) ⊃⋅⋅⋅ ⊃ em-1(A) ⊃ em(A) = em+1(A) = 0

becomes stable on a number m, where 0 is idempotent element.

Case (1)(a). If 0 6 j < m, then by Lemma 3.8 any quasigroup (Dj , o) is a right loop, isotope of a group (Dj ,+) of the form (Dj , o) = (Dj , +)(ε, , ε), where ∈ Aut(Dj ,+).

Case (1)(b). \Behaviourε of the map f in the right loop (Dj , o) is described by Lemma 3.9. If f is zero endomorphism, then (Dj , o) is a group in case j = 0 (Lemma 3.9(i)) and it is an Abelian group in the case j > 0 (Lemma 3.9(ii)).

If f is a nonzero endomorphism of (Dj , o), then information on the structure of (Dj , o) follows from Lemma 3.9 and Corollary 3.13.

Case (1)(c). The proof follows from Lemma 3.9(ii){(iv) and the fact that in the quasigroup Ker(fj j(Dj ,o)) the maps e and f are zero endomorphisms.

Case (1)(d). The proof follows from Corollary 3.13(1).

Case (1)(e). The proof follows from results of the previous cases of this theorem and Theorem 2.7(2).

Using Lemma 3.15 we can state that F-quasigroup (B,⋅) is isotopic to left distributive quasigroup (B,?), where x ? y = x⋅e(y).

In order to have more detailed information on the structure of the quasigroup em (Q,⋅) we study the following chain:

B ⊃ f(B) ⊃⋅⋅⋅⋅ ⊃ fr(B) = fr+1(B)

which becomes stable on a number r.

Case (2)(a). The proof follows from Corollary 3.13(2).

Case (2)(b). The proof follows from Theorem 3.17.

Case (2)(c). Since f is zero endomorphism of quasigroup (Hj,⋅), ej | Hj is a permutation of the set Hj , then by Corollary 3.13 quasigroup (Hj,⋅) has the form x⋅y = φx + y, where (Hj,+) is an Abelian group, φ ∈ Aut(Hj,+) and (Hj,o), x o y = φx + ey, is a medial distributive quasigroup.

Case (2)(d). The existence of direct decomposition follows from Theorem 2.7(2).

We notice that information on the structure of nite medial quasigroups is there in [103].

E-quasigroups

We recall a quasigroup (Q,⋅) is trimedial if and only if (Q,⋅) is an E-quasigroup [64]. Any trimedial quasigroup is isotopic to CML [50]. The structure of trimedial quasigroups has been studied in [20,54,107,59]. Here slightly other point of view on the structure of trimedial quasigroups is presented.

Lemma 3.24. (1) If endomorphism f of an E-quasigroup (Q,⋅) is zero endomorphism, i.e., f(x) = 0 for all x ∈ Q, then x⋅y = φx + y, (Q, +) is an Abelian group, φ ∈ Aut(Q, +).

(2) If endomorphism e of an E-quasigroup (Q,⋅) is zero endomorphism, i.e., e(x) = 0 for all x ∈ Q, then x⋅y = x + y, (Q, +) is an Abelian group, ∈ Aut(Q, +).

Proof. (1) from Theorem 2.14(3) it follows that (Q,⋅) is a left loop, x⋅y = αx + y, (Q,+) is an Abelian group, α ∈ SQ, α 0 = 0.

Further we have x  e(x) = αx + e(x) = x, αx = x - e(x) = (ε - e)x. Therefore α is an endomorphism of (Q, +), moreover, it is an automorphism of (Q, +), since α is a permutation of the set Q.

(2) The proof of Case (2) is similar to the proof of Case (1).

Corollary 3.25. If endomorphisms f and e of an E-quasigroup (Q,⋅) are zero endomor- phisms, i.e., f(x)=e(x)=0 for all x ∈ Q, then x⋅y=x + y, (Q, +) is an Abelian group.

Proof. from equality αx + e(x) = x of Lemma 3.24 we have αx = x, α = ε.

Corollary 3.26. (1) If endomorphism f of an E-quasigroup (Q,⋅) is zero endomorphism and endomorphism e is a permutation of the set Q, then x⋅y = φx+y, (Q, +) is an Abelian group, φ ∈ Aut(Q, +) and (Q, o), x o y = φx + ey, is a medial distributive quasigroup.

(2) If endomorphism e of an E-quasigroup (Q,⋅) is zero endomorphism and endomorphism f is a permutation of the set Q, then xy = x+ y, (Q, +) is an Abelian group, ∈ Aut(Q, +) and (Q, o), x o y = fx + y, is a medial distributive quasigroup.

Proof. (1) from Lemma 3.24 it follows that in this case (Q,⋅) has the form x⋅y = φx + y over Abelian group (Q, +). Then x  e(x) = φx + e(x) = x, e(x) = x - φx, e(0) = 0. We can rewrite equality e(x⋅y) = e(x)⋅e(y) in the form e(φx + y) = φe(x) + e(y). By y = 0 we have eφ(x) = φe(x). Then e(φx + y) = eφx + ey, the map e is an endomorphism of (Q,+). Moreover, the map e is an automorphism of (Q,+).

from Toyoda Theorem and equality e(x) = x - φx it follows that quasigroup (Q, o) is medial idempotent. It is well known that a medial idempotent quasigroup is distributive. Case (2) is proved in a similar way to Case (1).

Theorem 3.27. If the endomorphisms f and e of an E-quasigroup (Q,⋅) are permutations of the set Q, then quasigroup (Q, o) of the form x o y = f(x)⋅y is a distributive quasigroup and f, e ∈ Aut(Q, o).

Proof. The proof of this theorem is similar to the proof of Theorem 3.19.

By Lemma 2.15 (Q,⋅) is isotope of the form xy = f-1xoy of a left distributive quasigroup (Q, o) and f ∈ Aut(Q, o).

Moreover, by Lemma 2.15 (Q,⋅) is isotope of the form xy = xe-1y of a right distributive quasigroup and e ∈ Aut(Q, ). Therefore f-1x o y = x  e-1y, x o y = fx  e-1y.

Automorphisms e, f of the quasigroup (Q,⋅) lie in Aut(Q, o) (Lemma 1.24 or [72, Corollary 12]). We recall, ef = fe (Lemma 1.60).

Now we need to rewrite right distributive identity in terms of operation o. We have+

equation

If in the last equality we change element f2x by element x, element fe-1y = e-1fy by element y, element e-1z by element z, then we obtain

equation

In order to nish this proof we will apply Corollary 3.18.

Corollary 3.28. An E-quasigroup (Q,⋅) is simple if and only if this quasigroup lies in one from the following quasigroup classes:

(i) (Q,⋅) is a simple Abelian group in the case when the maps e and f are zero endomor- phisms,

(ii) (Q,⋅) is a simple medial quasigroup of the form x⋅y = φx + y in the case when the map f is a zero endomorphism and the map e is a permutation,

(iii) (Q,⋅) is a simple medial quasigroup of the form x⋅y = x + y in the case when the map e is a zero endomorphism and the map f is a permutation,

(iv) (Q,⋅) has the form x⋅y = x o y, where (Q, o) is a -simple distributive quasigroup, ∈ Aut(Q, o), in the case when the maps e and f are permutations.

Proof. (⇒) (i) The proof follows from Corollary 3.25. (ii) The proof follows from Lemma 3.24(1). (iii) The proof follows from Lemma 3.24(2). (iv) The proof is similar to the proof of Theorem 3.20(iv).

(⇐) It is clear that any quasigroup from these quasigroup classes is simple E-quasigroup.

Corollary 3.29. Finite simple E-quasigroup (Q,⋅) is a simple medial quasigroup.

Proof. The proof follows from Corollary 3.28 and is similar to the proof of Corollary 3.22. We can use Lemma 3.3.

Taking into consideration Corollary 3.29 we can say that properties of nite simple Equasigroups are described by Theorem 3.1.

Lemma 3.30. (1) If endomorphism f of an E-quasigroup (Q,⋅) is zero endomorphism, then equation where (A, o) a medial E-quasigroup of the form x⋅y = φx + y and there exists a number m such that jem(A, o)j = 1, (B,⋅) is a medial E-quasigroup of the form x⋅y = φx + y such that (B, ?), x ? y = φx + ey, is a medial distributive quasigroup.

(2) If endomorphism e of an E-quasigroup (Q,⋅) is zero endomorphism, then (Q,⋅) equation (A, +) × (B,⋅), where (A, o) is a medial E-quasigroup of the form x⋅y = x +ψy and there exists a number m such that jfm(A, o)j = 1, (B,⋅) is a medial E-quasigroup of the form x⋅y = x +ψy such that (B, ?), x ? y = fx +ψy, is a medial distributive quasigroup.

Proof. (1) By Theorem 2.17(4) any right E-quasigroup ((Q;.)) has the structure ((Q;.)) equation equationis a quasigroup with a unique idempotent element and there exists a number m such that equationis an isotope of a right distributive quasigroup equation

From Lemma 3.24 it follows thatequation has the form equationover an Abelian group (Q; +).

We recall that equation (Corollary 3.26). From equalities equation and equation Thenequation ) is medial, idempotent, therefore it is distributive.

(2) The proof is similar to Case (1).

Remark 3.31. If m = 1, then equation is an Abelian group (Corollary 3.25).

If m = 2, then equation s an extension of an Abelian group by an Abelian group. If, in addition, the conditions of Lemma 1.81 are ful lled, then equation is an Abelian group.

If the number m is nite and the conditions of Lemma 1.81 are ful lled, then after application of Lemma 1.81 equationtimes we obtain that equation is an Abelian group.

Now we have a possibility to give in more details information on the structure of finite E-quasigroups. The proof of the following theorem in many details is similar to the proof of Theorem 3.23.

Let Di be an equivalence class of the normal congruence Ker eicontaining an idempotent element a ∈ A, i ≥ 0. Let Hj be an equivalence class of the normal congruence Ker fj containing an idempotent element, j ≥ 0.

Theorem 3.32. In any nite E-quasigroup (Q; .) there exist the following nite chains:

equation

where

(1) if equation where right loop loop (Hi; +) is an extension of an Abelian group by Abelian groups, (Gi; ) is a medial E-quasigroup of the form equationsuch that is complete automorphism of the group (Gi; +);

(2) if i = m, then (emQ;.) is isotope of right distributive quasigroup equationwhere equationequation

(a) if j < r, then (Hj ; .) is medial left loop, (Hj ; ) has the formequation where (Hj ; +) is an Abelian group,equationequationis a medial distributive quasigroup;

(b) if j = r, then equationis isotope of the formequation of a distributive quasigroup equation

Proof. It is clear that in E-quasigroup (Q; .) chain (2.2) equation becomes stable.

(1) (i < m). By Lemma 3.24(2) any quasigroup (Di; ) is a medial right loop, isotope of an Abelian group (Di; +) of the form (Di;.) =equation for all suitable values of index i, since in the quasigroup (Di; .) endomorphism e is zero endomorphism.

If f is zero endomorphism, then in this case (Di; .) is an Abelian group (Corollary 3.25).

If f is a nonzero endomorphism of (Di; .), then we can use Lemma 3.30(2).

Case (1) (I). From Lemma 2.15(4) it follows that E-quasigroupequationis isotopic to right distributive quasigroup equationwhere equation

In order to have more detailed information on the structure of the quasigroup em(Q;.) we study the following chain:

equation

Case (2)(a) (j < r). From Lemma 2.15(4) it follows that E-quasigroup (Hj ;.) is isotopic to right distributive quasigroupequation

From Lemma 3.24(1) it follows that (Hj ;.) has the formequationwhere (Hj ; +) is an Abelian group,equation

From equalities equation we haveequation Then right distributive quasigroupequation is isotopic to Abelian group (Hj ; +)

If we rewrite identity equation in terms of the operation +, then equation By z = 0 from the last equality it follows that equation Thenequation is a medial quasigroup. Moreover, equation is a medial distributive quasigroup, since any medial right distributive quasigroup is distributive.

Case (2)(b) (j = r). If e and f are permutations of the set equationthen by Theorem 3.27 equation is isotope of the formequation of a distributive quasigroupequation

SM-quasigroups

We recall that left and right SM-quasigroup is called an SM-quasigroup. The structure theory of SM-quasigroups mainly has been developed by Kepka and Shchukin [52, 51, 106, 6].

If an SM-quasigroup (Q;.) is simple, then the endomorphism s is zero endomorphism or a permutation of the set Q.

If s(x) = 0, then from Theorem 2.14 we have the following.

Corollary 3.33. If the endomorphism s of a semimedial quasigroup ((Q;.)) is zero endomor phism, i.e., s(x) = 0 for all equation is a medial unipotent quasigroup,equationequation is an Abelian group,equation

Remark 3.34. By Corollary 3.33 equivalence class Di of the congruence Ker si containing an idempotent element is a medial unipotent quasigroup (Di;.) of the form equationwhere (Di; +) is an Abelian group, equation for all suitable values of index i.

Information on the structure of medial unipotent quasigroups is there in [103]. If s(x) is a permutation of the set Q, then from Lemma 2.15 we have the following.

Lemma 3.35. If the endomorphism s of a semimedial quasigroup ((Q;.)) is a permutation of the set Q, then quasigroup equation of the formequation is a distributive quasigroup and equation

Corollary 3.36. Any semimedial quasigroup (Q; .) has the structure equation whereequationis a quasigroup with a unique idempotent element and there exists a number m such that equation is an isotope of a distributive quasigroupequation for allequation

Proof. The proof follows from Theorem 2.17(3), (4).

Corollary 3.37. An SM-quasigroup ((Q;.)) is simple if and only if it lies in one from the following quasigroup classes:

(i) ((Q;.)) is a medial unipotent quasigroup of the formequation is Abelian group, equation and the group (Q; +) is equation

(ii) ((Q;.)) has the form equation is equationdistributive quasigroup.

Proof. The proof follows from Theorem 3.4(3), (4).

The similar result on properties of simple SM-quasigroups is there in [106, Corollary 4.13].

Corollary 3.38. Any nite simple semimedial quasigroup (Q; .) is a simple medial quasi- group [106].

Proof. Conditions of Lemma 3.3 are ful lled and we can apply it.

Simple left FESM-quasigroups

Kinyon and Phillips have de ned left FESM-quasigroups in [64].

De nition 3.39. A quasigroup ((Q;.)) which simultaneously is left F-, E-, and SM-quasigroup we will name left FESM-quasigroup. From De nition 3.39 it follows that in FESM-quasigroup the aps iare its endomorphisms.

Lemma 3.40. (1) If endomorphism e of a left FESM-quasigroup ((Q;.)) is zero endomor- phism, then ((Q;.)) is a medial right loop, equationis an Abelian group, equationAut equation

(2) If endomorphism f of an FESM-quasigroup ((Q;.)) is zero endomorphism, then (Q; .) is a medial left loop, i.e.,equation is an Abelian group,equationequation

(3) If endomorphism s of a left FESM-quasigroup ((Q;.)) is zero endomorphism, then (Q;.) is medial unipotent quasigroup of the form equation is an Abelian group,equation

Proof. (1) From Lemma 2.2 it follows that ((Q;.)) has the form equationis a group equation

Then equation Since s is an endomorphism of ((Q;.)), further we have equationequationequationTherefore equationthe group (Q; +) is commutative.

From equality equation

Further we have equation

(2) From Theorem 2.14(3) it follows that (Q; .) is a left loop equation an Abelian group,equation

Further we have equationTherefore ' is an endomorphism of (Q; +), moreover, it is an automorphism of (Q; +), since equation is a permutation of the set Q.

Then equationFrom equality equationsx.

Further, equationThen equation

(3) From Theorem 2.14(1) it follows that ((Q;.)) is unipotent quasigroup of the form x.y = equationwhere (Q; +) is a group equation

Since f is an endomorphism of quasigroup equationequationIf x = 0, then equationThen f is an endomorphism of the group (Q; +). Similarly, equationis an endomorphism of the group (Q; +).

From equation we haveequation Thenequation(3.13)

Comparing the right sides of equalities (3.13) we obtain that (Q; +) is a commutative group.

Lemma 3.41. If endomorphisms e, f, and s of a left FESM-quasigroup ((Q;.)) are permu- tations of the set Q, then quasigroup equation of the form equationis a left distributive quasigroup and equation

Proof. By Lemma 2.3 endomorphism e of a left F-quasigroup ((Q;.)) is a permutation of the set Q if and only if quasigroup equation of the form equationis a left distributive quasigroup and equation [15]. Then equation

The fact that equation follows from Lemma 1.24(2).

Theorem 3.42. If ((Q;.)) is a simple left FESM-quasigroup, then

(i) ((Q;.)) is simple medial quasigroup in the case when at least one from the maps e, f, and s is zero endomorphism;

(ii) ((Q;.)) has the form equationis a -simple left distributive quasigroup, equationin the case when the maps e, f and s are permutations; in this case equation

Proof. It is possible to use Lemma 3.40 for the proof of Case (i) and Lemma 3.41 for the proof of Case (ii).

Example 3.43. By (Z7; +) we denote cyclic group of order 7 and we take Z7= {0; 1; 2; 3; 4; 5; 6}.

Quasigroup equationwhere equation, is simple medial FESM-quasigroup in which the maps e and s are zero endomorphisms, the map f is a permutation of the set Z7equation

Quasigroup equationis simple medial FESM-quasigroup in which endomorphisms e, f, s are permutations of the set Z7

Loop isotopes

In this section we give some results on the loops and left loops which are isotopic to left F-, SM-, E-, and FESM-quasigroups.

We recall that any F-quasigroup is isotopic to a Moufang loop [55, 57], any SM-quasigroup is isotopic to a commutative Moufang loop [51]. Since any E-quasigroup is an SM-quasigroup [52, 64], then any E-quasigroup also is isotopic to a commutative Moufang loop.

Left F-quasigroups

Taking into consideration Theorems 2.7 and 2.17, Lemma 1.79, and Corollary 1.80 we can study loop isotopes of the factors of direct decompositions of left and right F- and Equasigroups.

Theorem 4.1. (1) A left F-quasigroup ((Q;.)) is isotopic to the direct product of a group (A; +) and a left S-loop equation

(2) A right F-quasigroup ((Q;.)) is isotopic to the direct product of a group (A; +) and right S-loop equation

Proof. (1) By Theorem 2.7(1) any left F-quasigroup ((Q;.)) has the structure equation(B; ), where (A; º) is a quasigroup with a unique idempotent element; (B; ) is isotope of a left distributive quasigroup

equation

By Corollary 1.80, if a quasigroup Q is the direct product of quasigroups A and B, then there exists an isotopy equation of Q such thatequation Therefore we have a possibility to divide our proof into two steps.

Step 1. Denote a unique idempotent element of equation by 0. We notice thatequation Indeed, fromequation

From left F-equality equation we haveequation Thenequation

Consider isotope equation of the quasigroupequation We notice that equation is a left loop. IndeedequationFurther we have equationequation

Prove that equation From equalityequationequationIf we pass in the left F-equality to the operation ⊕, then we obtain equationequation If we changeequationthen we obtain equation(4.1)

Then equationis a left F-quasigroup with the left identity element. For short below in this theorem we will use denotation e instead of equation Further we pass from the operation ⊕ to the operation equationequationThen equation if and only ifequationequation

We express the map e(x) in terms of the operation +. We have equation Thenequation

If we denote the map equationby equationWe can rewrite (4.1) in terms of the loop operation + as follows:

equation(4.2)

From equation By y = 0 from the last equality
we have equation(4.3).

Thereforeequatione is a normal endomorphism of (A; +). Changing x by x and taking into consideration (4.3) we obtain from equality (4.2) the following equality:

equation(4.4)

Next part of the proof was obtained using Prover 9 which is developed by Professor W. McCune [75].

If we put in equality (4.4) y = z = 0, then αx + ex = x, or, equivalently,

equation(4.5)

If we put in equality (4.4) y = 0, then

equation(4.6)

If we apply equality (4.5) to equality (4.6), then

equation(4.7)

If we apply equality (4.5) to equality (4.4), then

equation(4.8)

If we change in equality (4.7) x by x + y, then

equation(4.9)

Taking into consideration Lemma 2.5 and Theorem 2.7 we can say that there exists a minimal number i ( finite or infinite) such that equationfor any equation

If we change in equality (4.4) equation then

equation(4.10)

If we change in (4.10) equation(equality 4.5), then

equation(4.11)

We change in equality (4.9) equationthen

equation(4.12)

We rewrite the left-hand side of equality (4.12) as follows:

equation(4.13)

From (4.12) and (4.13) we have

equation    (4.14)

We change in equality (4.9) x by equation, then

equation    (4.15)

We rewrite the left-hand side of equality (4.15) as follows:

equation    (4.16)

From (4.15) and (4.16) we have

equation    (4.17)

Begin Cycle

We change in equality (4.9) x by equation. Then

equation    (4.18)

We rewrite the left-hand side of equality (4.18) as follows:

equation    (4.19)

From (4.18) and (4.19) we have

equation

End Cycle

Therefore

equation

for any natural number i. If the number n is finite, then repeating Cycle necessary number of times we will obtain that x + (y + z) = (x + y) + z for all x; y; equation.

Since n is a fixed number (maybe and an infinite), then equation, where equation.

We can apply Cycle necessary number of times to obtain associativity. Indeed, suppose that λ is a minimal number such that

equation    (4.20)

and there exist equation such that

equation

But from the other side, if we apply Cycle to equality (4.20), then we obtain that

equation

for all equation i.e., λ is not a minimal number with declared properties.

Therefore our supposition is not true and

equation

for all suitable λ and all equation.

Step 2. From Theorems 2.7 and 1.33 it follows that

equation

is a left S-loop.

(2) This case is proved similarly to Case (1).

Corollary 4.2. A loop (Q; *), which is the direct product of a group (A; +) and a left S-loop equation, is a left special loop.

Proof. Indeed, any group is left special. Any left S-loop also is a special loop (see [81, p. 61]). Therefore (Q; *) is a left special loop.

Lemma 4.3. The fulfilment of equality (4.4) in the group (A; +) is equivalent to the fact that the triple equation is an autotopy of (A; +) for all equation.

Proof. From (4.4) by y = 0 we have

equation    (4.21)

i.e., equation.

If we change in (4.4) y by equation, then

equation    (4.22)

Equality (4.22) means that the group (A; +) has an autotopy of the form

equation

for all equation. Taking into consideration that equation, we can rewrite Tx in the form

equation

Corollary 4.4. If the group (A; +) has the property equation for all equation, then

equation

equation

equation

Proof. (i) It is well known that any autotopy of a group (A; +) has the form (Laδ,Rbδ,LaRbδ), where La is a left translation of the group (A; +), Rb is a right translation of this group, δ is an automorphism of this group [15].

Therefore if the triple Td is an autotopy of the loop (A; +), then we have

equation    (4.23)

Then Le(d)0 = Rbδ0, e(d) = b. From Ld0 = LaRbδ0 we have d = a + b, d = a + e(d). But d = αd + e(d). Therefore, a = αd.

We can rewrite equalities (4.23) in the form

equation    (4.24)

Then

equation

We notice that all permutations of the form equation form a subgroup H' of the group RM(A; +), since e is an endomorphism of the group (A; +).

By our assumption H' ⊆ LM(A; +). Then

equation

But equation [43,90]. Therefore equation for all equation, equation.

(ii) From (i) it follows that the triple equation is an autotopy of (A; +). Indeed, equality equation is true for all b, equation since equation.

Then the triple equation is an autotopy of (A; +), i.e., equation.

By z = 0 we have equation Then the triple equation is a loop autotopy.

The equality equation means that equation for all b, equation. If we change equation, then equation for all b, equation, equation.

(iii) From equality equation we have equation.

Remark 4.5. Conditions equation for all equation and equation are equivalent.

Corollary 4.6. If e(x) = 0 for all equation, then equation.

Proof. In this case equality (4.22) takes the form equation. If autotopy of such form true in a loop, then equation.

Sokhatsky has proved the following theorem (see [113, Theorem 17]).

Theorem 4.7. A group isotope (Q,·) with the form x.y = equation is a left F-quasigroup if and only if β is an automorphism of the group (Q; +), β commutes with and satisfies the identity equation.

Example 4.8. Dihedral group (D8, +) with the Cayley table

equation

has an endomorphism

equation

and permutation equation such that equation. Using this permutation and taking into consideration Theorem 4.7 we may construct left F-quasigroups (D8,.) and (D8,*) with the forms equation where β = (15)(24). These quasigroups are right-linear group isotopes but they are not left linear quasigroups equation. This example was constructed using Mace 4 [74].

Corollary 4.9. A left special loop equation is isotope of a left F-quasigroup (Q,·) if and only if equation is isotopic to the direct product of a group (A; +) and a left S-loop equation.

Proof. If a left special loop equation is an isotope of a left F-quasigroup (Q,·), then from Theorem 4.1 it follows that equation is isotopic to a loop (Q,*) which is the direct product of a group (A; +) and a left S-loop equation.

Conversely, suppose that a left special loop equation is an isotope a loop (Q,*) which is the direct product of a group (A; +) and a left S-loop equation. It is easy to see that isotopic image of group (A; +) of the form equation, where equation, is a left F-quasigroup.

From Theorem 1.33 we have that isotopic image of the loop equation of the form equation, where equation is complete automorphism of equation, is a left distributive quasigroup (B,O). By Lemma 2.3 (see also [15]) isotope of the form equation, where equation, is a left F-quasigroup. Hence, among isotopic images of the left special loop equation there exists a left F-quasigroup.

Corollary 4.9 gives an answer to Belousov 1a Problem [12].

Corollary 4.10. If (Q,*) is a left M-loop which is isotopic to a left F-quasigroup (Q,·), then (Q,*) is isotopic to the direct product of a group and LP-isotope of a left S-loop.

Proof. By Theorem 4.1 any left F-quasigroup (Q,·) is LP-isotopic to a loop equation which is the direct product of a group equation and left S-loop equation.

By Theorem 1.35 any loop which is isotopic to a left F-quasigroup is a left M-loop.

Up to isomorphism (Q,*) is an LP-isotope of (Q,·). Then the loops (Q,*) and equation are isotopic with an isotopy equation. Moreover, they are LP-isotopic (see [15, Lemma 1.1]).

From the proof of Lemma 1.79 it follows that LP-isotopic image of a loop that is a direct product of two subloops also is isomorphic to the direct product of some subloops.

By Albert Theorem (Theorem 1.36) LP-isotopic image of a group is a group.

F-quasigroups

Theorem 4.11. Any F-quasigroup (Q,·) is isotopic to the direct product of a group equation equation(G, +) and a commutative Moufang loop equation i.e., (Q,·) equation.

Proof. By Theorem 2.7(1) any left F-quasigroup (Q,·) has the structure (Q,·) equation (B,.), where (A,0) is a quasigroup with a unique idempotent element; (B,.) is isotope of a left distributive quasigroup equation, equation for all x, equation.

By Theorem 2.7(2) the quasigroup (B,.) has the structure equation, where (G,0) is a quasigroup with a unique idempotent element; (K,.)is isotope of a right distributive quasigroup equation for all x, equation.

By Theorem 4.1(1), the quasigroup (A,0) is a group isotope. By Theorem 4.1(2), the quasigroup (G,0) is a group isotope.

In the quasigroup (K,.) the endomorphisms e and f are permutations of the set K and by Theorem 3.19 (K,.) is isotope of a distributive quasigroup. Then by Belousov Theorem (Theorem 1.31) quasigroup (K,.) is isotope of a CML equation.Therefore equation equation.

Theorem 4.12. Any loop (Q,*) that is isotopic to an F-quasigroup (Q,·) is isomorphic to the direct product of a group and a Moufang loop [55, 57].

Proof. By Theorem 4.11 an F-quasigroup (Q,·) is isotopic to a loop equation which is the direct product of a group and a commutative Moufang loop. Then any left translation L of (Q, +) is possible to be present as a pair (L1,L2), where L1 is a left translation of the loop (A,+), L2 is a left translation of the loop (B,+).

From Lemma 1.62 it follows that any LP-isotope of the loop (Q,+) is the direct product of its subloops.

By generalized Albert Theorem LP-isotope of a group is a group. Any LP-isotope of a commutative Moufang loop is a Moufang loop [12].

Corollary 4.13. If (Q,*) is an M-loop which is isotopic to an F-quasigroup, then (Q,*) is a Moufang loop.

Proof. The proof follows from Theorem 4.12. It is well known that any group is a Moufang loop.

Left SM-quasigroups

Theorem 4.14. A left SM-quasigroup (Q,·) is isotopic to the direct product of a group equation and a left S-loop equation, i.e., equation.

Proof. In many details the proof of this theorem repeats the proof of Theorem 4.1.

By Theorem 2.17 any left SM-quasigroup (Q,·) has the structure equation, where (A,o) is a quasigroup with a unique idempotent element and there exists a number m such that equation is an isotope of a left distributive quasigroup equation, equation.

By Corollary 1.80, if a quasigroup Q is the direct product of quasigroups A and B, then there exists an isotopy equation of Q such that equation is a loop.

Therefore we have a possibility to divide our proof into two steps.

Step 1. Denote a unique idempotent element of (A,o) by 0. It is easy to check that s° 0 = 0.

Indeed, from (s°)mA = 0 we have (s°)m+1A = s°0 = 0.

(23)-parastrophe of (A,o) is left F-quasigroup (A,.) (Lemma 1.85, (5)) such that |em(A,.)|=1. Then (A,.) also has a unique idempotent element. By Theorem 4.1 principal isotope of (A,.) is a group equation.

We will use multiplication of isostrophies (Definition 1.89, Corollary 1.90, and Lemma 1.91). (23)-parastrophe image of group equation coincides with its isotope of the form equation, where equation for all equation. if equation, then equation. But equation. Therefore equation, i.e., equation. Then equation, since equation.

We have

equation.

Step 2. The proof of this step is similar to the proof of Step 2 from Theorem 4.1 and we omit them.

Left E-quasigroups

Lemma 4.15. A left E-quasigroup (Q,·) is isotopic to the direct product of a left loop equation with equality equation, where δ is an endomorphism of the loop equation, and a left S-loop equation.

Proof. In some details the proof of Lemma 4.15 repeats the proof of Theorem 4.1. By Theorem 2.17 any left E-quasigroup (Q,·) has the structure equation, where(A,o) is a quasigroup with a unique idempotent element and there exists a number m such that equation is an isotope of a left distributive quasigroup equationequation for all equation.

By Corollary 1.80, if a quasigroup Q is the direct product of quasigroups A and B, then there exists an isotopy equation of Q such that equation is a loop.

Therefore we have a possibility to divide our proof into two steps.

Step 1. We will prove that equation is a left loop. Denote a unique idempotent element of (A,o) by 0. It is easy to check that f° 0 = 0. Indeed, from (f°)mA = 0 we have equationequation.

From left E-equality equation by x = 0 we have equation. Then equation.

Consider isotope equation of the quasigroup equation. We notice that equation is a left loop. Indeed, equation.

Prove that equation. From equality equation we have equationequation.

If we pass in left E-equality to the operation equation, then we obtain equationequation. If we change L0y by y, L02z by z, then we obtain

equation    (4.25)

We notice that equation. Then equation. Moreover, from equation we have equation. If x = 0, then equationequation is an endomorphism of the left loop equation.

We can rewrite equality (4.25) in the following form:

equation    (4.26)

If we change in (4.26) x by Lx-1, then we obtain

equation    (4.27)

If we denote the map equation of the set Q by δ, then from (4.27) we have equationequation. The map equation is an endomorphism of the left loop equation since f° is an endomorphism and L0-1 an automorphism of equation. We notice, equation.

Step 2. From Theorems 2.17 and 1.33 it follows that

equation is a left S-loop.

Remark 4.16. If we take equation, then from equation we have equation. Thus from(4.26) we have equation.

Lemma 4.17. A left E-quasigroup (Q,·) is isotopic to the direct product of a loop (A, +) with equality equation, where δ is an endomorphism of the loop (A, +), and a left S-loop equation.

Proof. We pass from the operation equation to operation equation. Then equation, where x/y=z if and only if equation We notice that equation, since equation.

If we denote the map equation by α, then equation. We can rewrite (4.27) in terms of the loop operation + as follows:

equation    (4.28)

Prove that equation. Notice that equation. Then equation. Thus

equation

Then δ is an endomorphism of the loop (A, +). Indeed, equationequation

Equality (4.28) takes the form

equation    (4.29)

If we put in equality (4.29) x = y, then equation and equality (4.29) takes the form

equation    (4.30)

Lemma 4.18. If equation for all equation, then (A, +) is a commutative group.

Proof. If we put in equality (4.30) z = 0, then x+y = y +x. Therefore, from x+(y +z) = y + (x + z) we have (y + z) + x = y + (z + x).

Lemma 4.19. There exists a number m such that in the loop (A, +) the chain equation    (4.31)

is stabilized on the element 0.

Proof. From Theorem 2.17 it follows that (A,0) is a left E-quasigroup with a unique idempotent element 0 such that the chain

equation    (4.32)

is stabilized on the element 0.

From Lemmas 4.15 and 4.17 it follows that (A, +) = (A,o)T, where isotopy T has the form equation. Since equation, then equation is a subloop of the loop (A, +) (Lemma 1.14) for all suitable values of i.

Thus we obtain that the isotopic image of chain (4.32) is the following chain:

equation    (4.33)

We recall that equation (Lemma 4.15). Then equation and equation. It is clear that equation is a bijection of the set A for all suitable values of i.

Thus we can establish the following bijection: equation. Then equationequation, since equation. Therefore equationequation.

Lemma 4.20. The loop (A,+) is a commutative group.

Proof. From Lemma 4.19 it follows that in (A, +) there exists a number m such that equation for all equation. We have used Prover's 9 help [75]. From (4.30) by y = 0 we obtain

equation    (4.34)

If we change in equality (4.34) y by y + z, then we obtain

equation    (4.35)

From (4.30) by z = 0 using (4.34) we have

equation    (4.36)

If we change in (4.36) y by equation, then

equation    (4.37)

But equation (Definition 1.15, equality (1.1)). Therefore

equation    (4.38)

If we change in (4.30) x by equation, then, using condition equation, we have

equation    (4.39)

Begin Cycle

If we change in equality (4.38) the element x by the element equation, then we have

equation    (4.40)

If we change in (4.39) z by equation, then, using Definition 1.15, equality (1.1), we obtain

equation    (4.41)

If we change in (4.41) y by equation, z by y and compare (4.41) with (4.40), then we obtain

equation    (4.42)

We have equation since δ is an endomorphism of the loop (A; +). Notice from equalities (4.39) and (4.42) it follows that equation.

From equality equation (Definition 1.15, equality (1.2)) using commutativity(4.42) we obtain

equation    (4.43)

From equality (4.43) and definition of the operation n we have

equation    (4.44)

If we change in (4.39) y + z by y, then y pass in y/z and we have

equation    (4.45)

Applying to (4.45) the operation / we have

equation    (4.46)

Write equality (4.30) in the form

equation    (4.47)

From (4.47) using (4.35) we obtain

equation    (4.48)

From equality (4.48) using (4.44) we have

equation    (4.49)

If we change in equality (4.49) x by equation, then we obtain

equation    (4.50)

Using equality (4.46) in equality (4.50) we have

equation    (4.51)

Therefore

equation

and

equation    (4.52)

End Cycle

Therefore we can change equality (4.39) by the equality (4.52) and start new step of the cycle.

After m steps we obtain that in the loop (A, +) the equality x + (y + z) = y + (x + z) is fulfilled, i.e., (A, +) is an Abelian group. If m = ∞, then we can use arguments similar to the arguments from the proof of Theorem 4.1.

Theorem 4.21. (1) A left E-quasigroup (Q,·) is isotopic to the direct product of an Abelian group (A,+) and a left S-loop equation.

(2) A right E-quasigroup (Q,·) is isotopic to the direct product of an Abelian group (A, +) and a right S-loop equation.

Proof. (1) The proof follows from Lemmas 4.17 and 4.18.

Theorem 4.21 gives an answer to Kinyon-Phillips problems (see [64, Problem 2.8, (1)]).

Corollary 4.22. A left FESM-quasigroup (Q,·) is isotopic to the direct product of an Abelian group equation and a left S-loop equation.

Proof. We can use Theorem 4.21.

Corollary 4.22 gives an answer to Kinyon-Phillips problem (see [64, Problem 2.8, (2)]). We hope in a forthcoming paper we will discuss a generalization of Murdoch theorems about the structure of nite binary and n-ary medial quasigroups [78,102] on in nite case and medial groupoids.

Acknowledgment

The author thanks MRDA-CRDF (ETGP, Grant no. 1133), Consiliul Suprem pentru equation equation al Republicii Moldova (Grant no. 08.820.08.08 RF) and organizers of the conference LOOPS'07 for nancial support. The author also thanks Professor V. I. Arnautov for his helpful comments.

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