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On the structure of left and right F-, SM-, and E-quasigroups | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications

# 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

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#### 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 , we can say that we study some quasigroups from the following quasigroup classes: .

This paper is connected with the following problems.

Problem 1 (Belousov Problem 1a [12,55,98]). Find necessary and sucient 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 . Describe quasigroups with the property f(ab) = f(a)f(b) for all , 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 be a groupoid (be a magma in alternative terminology). As usual, the map for all x ∈ Q, is a left translation of the groupoid relative to a xed element a ∈ Q; the map , is a right translation.

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

In a division groupoid , 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 is said to be a cancellation groupoid if , for all a, b, c ∈ G.

If any from equations a ⋅ x = b and y ⋅ a = b has a solution in a cancellation groupoid 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 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 is a bijection of the set Q.

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

A sub-object of a quasigroup 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 of the set Q such that for all. We also can say that a groupoid (Q,A) is an isotopic image of a groupoid (Q,B). The triple () is called an isotopy (isotopism).

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

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

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

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

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 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):

We will denote

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

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

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

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

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

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 is a quasigroup, then the map 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 are given. This table in fact is there in [13]; see also [31,94].

In Table 1, for example, .

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

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

Definition 1.10. An element f(b) of a quasigroup 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 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 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 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 means that e(x) = f(x) = e for all x ∈ Q, i.e., all left and right local identity elements in the quasigroup coincide [12].

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

In Table 2, for example,

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

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

Quasigroup isotopy of the form 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 a quasigroup. If (Q, +) = , then for some translations of .

Lemma 1.14. If is a quasigroup, 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 .

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

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

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

are fulfilled.

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

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

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

Proof. (1) If a, b, c ∈ H, then from a ⋅ b = a ⋅ c follows b = c, since . 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 are defined in the following way:

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

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 into itself, i.e., a triple (α, β, γ) of permutations of the set Q is an autotopy if the equality is fulfilled for all x, y ∈ Q.

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

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

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

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

Proof. Let T = (α, β, γ) be an autotopy of a loop , i.e.,. If we put x = 1, then we obtain. If we put y = 1, then, by analogy, we obtain, . Then, 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

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 , then . (2) If, then. (3) If, then a ∈ C(Q, +).

Proof. (3) We have .

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

Proof. We have , where.

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 , in particular,.

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

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

(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 for all

• left distributive, if for all

• right distributive, if for all

• distributive, if it is left and right distributive;

• idempotent, if for all

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

• left semi-symmetric, if for all

• TS-quasigroup, if for all

• left F-quasigroup, if for all

• right F-quasigroup, if for all

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

• right semimedial, if for all

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

• left E-quasigroup, if for all

• left E-quasigroup, if for all

• right E-quasigroup, if for all

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

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

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

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

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

Definition 1.26. A loop

Bol loop (left Bol loop), if for all

Moufang loop, if for all

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

right M-loop, if for all where is
a mapping of the set Q;

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

• left special, if is an automorphism of for any pair

• right special, if is an automorphism of for any pair

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 the equalities take the form respectively, and they are identities in

Therefore any subquasigroup of a left F-quasigroup is a left F-quasigroup; any homomorphic image of a left F-quasigroup 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 is both a left and right F-, SM-, and E-quasigroup.

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

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

Proof. (1) It is easy to see that is idempotent quasigroup. ThereforeThen

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

Lemma 1.29. A quasigroup in which

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

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

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

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

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

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

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

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

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

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

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

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

The following theorem is proved in [17].

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

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

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

A left distributive quasigroup with identity 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 [21,71,82].

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

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

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

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

Definition 1.39. An equivalence is a congruence of a groupoid if the following implications are true for all

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

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

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

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

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

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

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

Moreover, we will say that a binary relation 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 be a normal congruence of a quasigroup Prove the following implication:

(1.5)

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

Implication

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

Corollary 1.46. If is a normal quasigroup congruence of a quasigroup Q, then 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

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

Since and is a normal quasigroup congruence, we have

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 and be normal congruences of a quasigroup Then means that there exists an element such that and

Further, we have

Then

From relations

we obtain

Therefore,

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

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

In case 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 is a division groupoid [5,24].

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

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

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

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

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

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

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

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 and then moreover since Conversely, let If then and Then

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

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

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

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

For equation the proof is similar.

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

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

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

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

(2) From Case (1) and Lemma 1.54 it follows that any endomorphic image of a left
F-quasigroup 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 then h is an endomorphism of the quasigroups i.e., from we
obtain that

Proof. From Lemma 1.54 we have that is a subquasigroup of

(1) If we pass from the quasigroup to quasigroup then subquasigroup of the quasigroup will correspond to the subquasigroup Indeed, any subquasigroup
of the quasigroup is closed relative to parastrophe operations of the
quasigroup Further we have

(2) If we pass from the quasigroup to quasigroup then subquasigroup of the quasigroup will correspond to the subquasigroup

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

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

Lemma 1.60. Let be a quasigroup.

• If f is an endomorphism of then for all

• If e is an endomorphism of then for all

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

Proof. We will use Corollary 1.58.

• If f is an endomorphism, then

• If e is an endomorphism, then

• If s is an endomorphism, then

This proves the lemma.

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

The group is called inner mapping group of a quasigroup (Q, .) relative to an element Group is stabilizer of a fixed element h by action on the set Q. In loop case usually it is studied the group 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 for a fixed element [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 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, where be a normal congruence of (Q, +). Then θ is normal congruence of (Q,.) if and only if 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 for all then

Proof. If θ is a normal congruence of a quasigroup (Q,.), then, since θ is admissible relative to the isotopy 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 there exists endomorphism and

Example 1.66. In the cyclic group (Z4, +), Z4 = {0; 1; 2; 3}, there exists endomorphism such that 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

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

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, +), where (Q, +) is a loop, αIf (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 is the set of all ordered pairs and where the operation in (Q,. *) is defined componentwise, that is,

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

If Ω is an -algebra and U, W are congruences on A, then , and 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 then the join and W is called direct product 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 if and only if there exist such congruences U and W of A that .

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

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 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., and B a subquasigroup of Q. A quasigroup Q is the semidirect product of quasigroups A and B, if there exists a homomorphism 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:

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, then there exists an isotopy T of Q such that QT is a loop and

Proof. If we take isotopy of the form , where a 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 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 is a loop.

Proof. The proof follows from Lemma 1.79.

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

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

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), where We notice since (K; ), (H; ) are subgroups of the left loop (Q,.). Indeed, from for all we have

Further we have

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

Example 1.82. Medial quasigroup where (Z9; +) is the cyclic group, 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 , 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 , then α 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 is an automorphism of this quasigroup. Indeed, we can rewrite left distributive identity in such manner Using Table 1 we have that Thus by Lemma 1.84 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

If We notice, if then See Table 2.

We can rewrite equality Now we have the equality If we denote then

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

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 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 are permutations of the set Q such that

for all

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

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 are isostrophisms of a quasigroup (Q;A), then

where for any quasigroup triplet [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 has

Lemma 1.91. One has

Proof. Let be an isotopy of a quasigroup Then

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 of binary quasigroup operation defined on a non- empty set Q is the general solution of the generalized associativity equation

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

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

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

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

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

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

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

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

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

Proof. (1) We have

(2) We have

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

(4)We have , where .

(5) If , then . Therefore and .

(6) If , then . Therefore and .

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

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

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

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

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

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 be a principal isotope of a group (Q, +), .

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

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

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

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

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

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

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 over a group (Q, +) the equality holds for all and fixed , then (Q, +) is an Abelian group.

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

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

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

Case (2) is proved in a similar way.

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

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

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

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

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 is quasigroup. Indeed, if z = c, then we have is isotope of quasigroup . Therefore is a quasigroup. Moreover, , where .

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

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

Quasigroup with generalized identity , where 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 fulfills Sushkevich postulate B, then is iso- topic to the group , where (see [15, Theorem 1.8]).

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

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

(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 fulfills postulates A* and B*, then by Theorem 1.100(2), groupoid (magma), is a group and . By the same theorem . Therefore is an autotopy of the group . By Corollary 1.22 . Therefore . It is easy to see that 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 usual, and so on.

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

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

Proof. (1) From identity we have , i.e., . Further we have and so on. Therefore is an endomorphism of the quasigroup . The fact that is a subquasigroup of quasigroup 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 is zero endomorphism, i.e., for all , if and only if left F-quasigroup is a right loop, isotope of a group (Q, +) of the form , where.

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

Proof. (1) We can rewrite equality in the form , where . Therefore Sushkevich postulate B is fulfilled in and we can apply Theorem 1.101. Further we have . From the other side . 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 is a permutation of the set Q if and only if quasigroup of the form is a left distributive quasigroup and .

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

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

(2.1)

Prove that . We have [72].

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

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

(2) The proof is similar.

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

(2.2)

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

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 is an automorphism of quasigroup .

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

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

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

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

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

Example 2.6. Quasigroup , where is in nite cyclic group, is medial, unipotent, left F-quasigroup such that . Notice in this case . 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 has the following structure:

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

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

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

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 is isotope of left distributive quasigroup.

If , where k is a fixed element of the set Q, then the quasigroup 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 , where m > 1.

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

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

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

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

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

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

(2.3)

then .

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

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

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

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

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

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

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

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

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

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

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

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

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

Left F-quasigroup equality holds in quasigroup since .

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

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

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

From properties of quasigroup we have that , 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 , we obtain . Therefore , i.e., the element a is an idempotent element of quasigroup .

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

The fact that is isotope of a left distributive quasigroup 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 . By we denote endomorphic image of the quasigroup relative to the endomorphism ej.

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

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

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

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

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

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

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

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 of the normal congruence Ker fj containing an idempotent element a Q forms linear left loop (; ) 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 Q, then (Q,. ) is an unipotent quasigroup, (Q, . ) , where , (Q, +) is a group, Aut(Q; +).

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

, (Q, +) is a group, 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 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 Q. If we denote xz by v, then and equality k. yz = xy. xz takes the form .

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, , 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, Aut(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 Q. Then α = Iψ , where x + I(x) = 0 for all x Q. Therefore α is an antiautomorphism of the group (Q, +), x.y = Iψ x + ψy + k.

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

Below we will suppose that any left semimedial quasigroup (Q, . ) with zero endomorphism s is an unipotent quasigroup with the form , where (Q, +) is a group, ' 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 and the equality zy.k = zx.yx takes the form

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 = , 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, 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 Q. Then β = Iψ, where Ix + x = 0 for all x Q.

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

We have . It is easy to see that for all x 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, 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 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 Q. Then ψ = ε x.y = αx + y + c = Lcαx + y for all x; y Q. In other words, x.y = αx + y for all x; y Q.

Further let a + α0 = 0. Then

(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 Q. Then β = ", x.y = x + βy + c = x + Rcβy for all x; y Q. In other words, x.y = x + βy for all x; y Q.

Further let a + β0 = 0. Then

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 of the form is a left distributive quasigroup and s Aut.

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

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

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

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

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

Prove that s Aut We have

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

Prove that f Aut . 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 Aut(Q, . ). We have

Prove that e Aut . 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:

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

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

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 B, s Aut(B, . ), s Aut(B;?).

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

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

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

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 B, e Aut(B, . ), e 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, . ) (Q, . ); if (Q, . ) is a right SM-quasigroup, then sm(Q, . )) P (Q, . ); if (Q, . ) is a left E-quasigroup, then fm(Q, . ) (Q, . ); if (Q, . ) is a right E-quasigroup, then em(Q, . ) (Q, . ).

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

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

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

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

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

(2) C(Q; +) ≅ × 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 (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 = Hj of the normal congruence Ker sj containing an idempotent element 0 Q is an unipotent loop (Hj ; . ) isotopic to

Since (Hj,.) is a commutative loop, we have that 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;) any nonzero element has the order 2i or infinite order.

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

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

Therefore (A,) ≅ 3(A,) ⊆ C(A,), (A,) 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,)*C(B, +). Therefore C(Q, +) ≅ (A,) * C(B, +) since C(A,) = (A,).

(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,)*(B, +))=((A,)*C(B, +)) follows from Cases (1) and (2).

Isomorphism

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

(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,) has the order 2i, 1 ≤ i ≤ m; (ii) Aut(Q, +) ≅ Aut(A,) * 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,) and (B, +) have elements of different orders. Indeed, the orders of elements of the loop (A,) are powers of the number 2 and orders of the elements of the loop (B, +) are some odd numbers or, possibly, ∞

Therefore loops (A,) and (B, +) are invariant relative to any automorphism of the loop (Q, +). Then Aut(Q, +) ≅ Aut(A,) * 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 Jezek-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 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 (see Remark 2.21), then we have thus .For any distributive quasigroup (Q, o) of the form we have

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

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 [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 where and the group (Q, +) is -simple;

(ii) (Q,.) has the form x.y = x o y, where ∈ Aut(Q; o) and (Q; o) is -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 (Q, +) is a group, ∈ Aut(Q, +) and the group (Q, +) is -simple;

(ii) (Q,. ) has the form x .y = (x 0 y), where ∈ Aut(Q, 0) and (Q, 0) is -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 = x o y, (Q, +) is a group, ∈ Aut(Q, +) and the group (Q,+) is -simple;

(ii) (Q,.) has the form x.y = (x o y), where ∈ Aut(Q, o) and (Q, o) is -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 = x o y, where ∈ Aut(Q, o) and (Q, o) is -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 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 = x + y, (Q, +) is a group, ∈ 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 If we take y = 0 in the last equality, then Therefore

Conversely, from x.y = x + y we have 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), 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; ( ; ), where 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)

(ii)e(Q, +) ⊆ C(Q, +);

(iii) 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; (;), where 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 We can rewrite equality f(x. y) = f(x) . f(y) in the form then we have f(0) = 0. If x = 0, then .Therefore

(3.1)

(ii) If we apply to equality (3.1) the equality then we obtain i.e.,

(iii) From de nitions and Case (ii) it follows that The 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).

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

(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 = x + y, (Q; +) is an Abelian group, ∈ Aut(Q, +) and (Q, o), x o y = x + 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 fxx = 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

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

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:

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

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

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

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

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

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

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:

where (A, o) is a quasigroup with a unique idempotent element, (B,⋅) is isotope of a left distributive quasigroup (B,?), xy = 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 xy = 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 xy = 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 xy = xe-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+

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

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 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,⋅) (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;.)) is a quasigroup with a unique idempotent element and there exists a number m such that is an isotope of a right distributive quasigroup

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

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

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

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

If m = 2, then s an extension of an Abelian group by an Abelian group. If, in addition, the conditions of Lemma 1.81 are ful lled, then 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 times we obtain that 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:

where

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

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

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

(b) if j = r, then is isotope of the form of a distributive quasigroup

Proof. It is clear that in E-quasigroup (Q; .) chain (2.2) 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;.) = 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-quasigroupis isotopic to right distributive quasigroup where

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

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

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

From equalities we have Then right distributive quasigroup is isotopic to Abelian group (Hj ; +)

If we rewrite identity in terms of the operation +, then By z = 0 from the last equality it follows that Then is a medial quasigroup. Moreover, 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 then by Theorem 3.27 is isotope of the form of a distributive quasigroup

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 is a medial unipotent quasigroup, is an Abelian group,

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 where (Di; +) is an Abelian group, 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 of the form is a distributive quasigroup and

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

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 form is Abelian group, and the group (Q; +) is

(ii) ((Q;.)) has the form is distributive 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, is an Abelian group, Aut

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

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

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

Then Since s is an endomorphism of ((Q;.)), further we have Therefore the group (Q; +) is commutative.

From equality

Further we have

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

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

Then From equality sx.

Further, Then

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

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

From we have Then(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 of the form is a left distributive quasigroup and

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 of the form is a left distributive quasigroup and [15]. Then

The fact that 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 is a -simple left distributive quasigroup, in the case when the maps e, f and s are permutations; in this case

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 where , is simple medial FESM-quasigroup in which the maps e and s are zero endomorphisms, the map f is a permutation of the set Z7

Quasigroup is 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

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

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

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

Step 1. Denote a unique idempotent element of by 0. We notice that Indeed, from

From left F-equality we have Then

Consider isotope of the quasigroup We notice that is a left loop. IndeedFurther we have

Prove that From equalityIf we pass in the left F-equality to the operation ⊕, then we obtain If we changethen we obtain (4.1)

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

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

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

(4.2)

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

Thereforee 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:

(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,

(4.5)

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

(4.6)

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

(4.7)

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

(4.8)

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

(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 for any

If we change in equality (4.4) then

(4.10)

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

(4.11)

We change in equality (4.9) then

(4.12)

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

(4.13)

From (4.12) and (4.13) we have

(4.14)

We change in equality (4.9) x by , then

(4.15)

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

(4.16)

From (4.15) and (4.16) we have

(4.17)

Begin Cycle

We change in equality (4.9) x by . Then

(4.18)

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

(4.19)

From (4.18) and (4.19) we have

End Cycle

Therefore

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; .

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

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

(4.20)

and there exist such that

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

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

Therefore our supposition is not true and

for all suitable λ and all .

Step 2. From Theorems 2.7 and 1.33 it follows that

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 , 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 is an autotopy of (A; +) for all .

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

(4.21)

i.e., .

If we change in (4.4) y by , then

(4.22)

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

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

Corollary 4.4. If the group (A; +) has the property for all , then

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

(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

(4.24)

Then

We notice that all permutations of the form 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

But [43,90]. Therefore for all , .

(ii) From (i) it follows that the triple is an autotopy of (A; +). Indeed, equality is true for all b, since .

Then the triple is an autotopy of (A; +), i.e., .

By z = 0 we have Then the triple is a loop autotopy.

The equality means that for all b, . If we change , then for all b, , .

(iii) From equality we have .

Remark 4.5. Conditions for all and are equivalent.

Corollary 4.6. If e(x) = 0 for all , then .

Proof. In this case equality (4.22) takes the form . If autotopy of such form true in a loop, then .

Sokhatsky has proved the following theorem (see [113, Theorem 17]).

Theorem 4.7. A group isotope (Q,·) with the form x.y = is a left F-quasigroup if and only if β is an automorphism of the group (Q; +), β commutes with and satisfies the identity .

Example 4.8. Dihedral group (D8, +) with the Cayley table

has an endomorphism

and permutation such that . Using this permutation and taking into consideration Theorem 4.7 we may construct left F-quasigroups (D8,.) and (D8,*) with the forms where β = (15)(24). These quasigroups are right-linear group isotopes but they are not left linear quasigroups . This example was constructed using Mace 4 [74].

Corollary 4.9. A left special loop is isotope of a left F-quasigroup (Q,·) if and only if is isotopic to the direct product of a group (A; +) and a left S-loop .

Proof. If a left special loop is an isotope of a left F-quasigroup (Q,·), then from Theorem 4.1 it follows that is isotopic to a loop (Q,*) which is the direct product of a group (A; +) and a left S-loop .

Conversely, suppose that a left special loop is an isotope a loop (Q,*) which is the direct product of a group (A; +) and a left S-loop . It is easy to see that isotopic image of group (A; +) of the form , where , is a left F-quasigroup.

From Theorem 1.33 we have that isotopic image of the loop of the form , where is complete automorphism of , is a left distributive quasigroup (B,O). By Lemma 2.3 (see also [15]) isotope of the form , where , is a left F-quasigroup. Hence, among isotopic images of the left special loop 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 which is the direct product of a group and left S-loop .

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 are isotopic with an isotopy . 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 (G, +) and a commutative Moufang loop i.e., (Q,·) .

Proof. By Theorem 2.7(1) any left F-quasigroup (Q,·) has the structure (Q,·) (B,.), where (A,0) is a quasigroup with a unique idempotent element; (B,.) is isotope of a left distributive quasigroup , for all x, .

By Theorem 2.7(2) the quasigroup (B,.) has the structure , where (G,0) is a quasigroup with a unique idempotent element; (K,.)is isotope of a right distributive quasigroup for all x, .

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 .Therefore .

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 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 and a left S-loop , i.e., .

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 , where (A,o) is a quasigroup with a unique idempotent element and there exists a number m such that is an isotope of a left distributive quasigroup , .

By Corollary 1.80, if a quasigroup Q is the direct product of quasigroups A and B, then there exists an isotopy of Q such that 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 .

We will use multiplication of isostrophies (Definition 1.89, Corollary 1.90, and Lemma 1.91). (23)-parastrophe image of group coincides with its isotope of the form , where for all . if , then . But . Therefore , i.e., . Then , since .

We have

.

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 with equality , where δ is an endomorphism of the loop , and a left S-loop .

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 , where(A,o) is a quasigroup with a unique idempotent element and there exists a number m such that is an isotope of a left distributive quasigroup for all .

By Corollary 1.80, if a quasigroup Q is the direct product of quasigroups A and B, then there exists an isotopy of Q such that is a loop.

Therefore we have a possibility to divide our proof into two steps.

Step 1. We will prove that 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 .

From left E-equality by x = 0 we have . Then .

Consider isotope of the quasigroup . We notice that is a left loop. Indeed, .

Prove that . From equality we have .

If we pass in left E-equality to the operation , then we obtain . If we change L0y by y, L02z by z, then we obtain

(4.25)

We notice that . Then . Moreover, from we have . If x = 0, then is an endomorphism of the left loop .

We can rewrite equality (4.25) in the following form:

(4.26)

If we change in (4.26) x by Lx-1, then we obtain

(4.27)

If we denote the map of the set Q by δ, then from (4.27) we have . The map is an endomorphism of the left loop since f° is an endomorphism and L0-1 an automorphism of . We notice, .

Step 2. From Theorems 2.17 and 1.33 it follows that

is a left S-loop.

Remark 4.16. If we take , then from we have . Thus from(4.26) we have .

Lemma 4.17. A left E-quasigroup (Q,·) is isotopic to the direct product of a loop (A, +) with equality , where δ is an endomorphism of the loop (A, +), and a left S-loop .

Proof. We pass from the operation to operation . Then , where x/y=z if and only if We notice that , since .

If we denote the map by α, then . We can rewrite (4.27) in terms of the loop operation + as follows:

(4.28)

Prove that . Notice that . Then . Thus

Then δ is an endomorphism of the loop (A, +). Indeed,

Equality (4.28) takes the form

(4.29)

If we put in equality (4.29) x = y, then and equality (4.29) takes the form

(4.30)

Lemma 4.18. If for all , 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     (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

(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 . Since , then 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:

(4.33)

We recall that (Lemma 4.15). Then and . It is clear that is a bijection of the set A for all suitable values of i.

Thus we can establish the following bijection: . Then , since . Therefore .

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 for all . We have used Prover's 9 help [75]. From (4.30) by y = 0 we obtain

(4.34)

If we change in equality (4.34) y by y + z, then we obtain

(4.35)

From (4.30) by z = 0 using (4.34) we have

(4.36)

If we change in (4.36) y by , then

(4.37)

But (Definition 1.15, equality (1.1)). Therefore

(4.38)

If we change in (4.30) x by , then, using condition , we have

(4.39)

Begin Cycle

If we change in equality (4.38) the element x by the element , then we have

(4.40)

If we change in (4.39) z by , then, using Definition 1.15, equality (1.1), we obtain

(4.41)

If we change in (4.41) y by , z by y and compare (4.41) with (4.40), then we obtain

(4.42)

We have since δ is an endomorphism of the loop (A; +). Notice from equalities (4.39) and (4.42) it follows that .

From equality (Definition 1.15, equality (1.2)) using commutativity(4.42) we obtain

(4.43)

From equality (4.43) and definition of the operation n we have

(4.44)

If we change in (4.39) y + z by y, then y pass in y/z and we have

(4.45)

Applying to (4.45) the operation / we have

(4.46)

Write equality (4.30) in the form

(4.47)

From (4.47) using (4.35) we obtain

(4.48)

From equality (4.48) using (4.44) we have

(4.49)

If we change in equality (4.49) x by , then we obtain

(4.50)

Using equality (4.46) in equality (4.50) we have

(4.51)

Therefore

and

(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 .

(2) A right E-quasigroup (Q,·) is isotopic to the direct product of an Abelian group (A, +) and a right S-loop .

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 and a left S-loop .

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 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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