Cheban loops

Left Cheban loops are loops that satisfy the identity x(xy.z) = yx.xz. Right Cheban loops satisfy the mirror identity {(z.yx)x = zx.xy}. Loops that are both left and right Cheban are called Cheban loops. Cheban loops can also be characterized as those loops that satisfy the identity x(xy.z) = (y.zx)x. These loops were introduced in Cheban, A. M. Loops with identities of length four and of rank three. II. (Russian) General algebra and discrete geometry, pp. 117-120, 164,"Shtiintsa", Kishinev, 1980. Here we initiate a study of their structural properties. Left Cheban loops are left conjugacy closed. Cheban loops are weak inverse property, power associative, conjugacy closed loops; they are centrally nilpotent of class at most two.


Introduction
In [15], Marshall Osborn showed that if L is a weak inverse property loop all of whose loop isotopes also have the weak inverse property, then the nucleus of L is normal and the quotient of L by its nucleus is Moufang (definitions are given in the next section). He showed further that L must satisfy the identity x(yz · x) = ((x · yx)/x) · zx. In [1], A.S. Basarab (Basarab) coined the term Osborn loop to describe those loops that satisfy this identity.
The variety of Osborn loops contains as subvarieties two of the most important classes of loops-the Moufang loops and the conjugacy closed loops [12]. One of the most important open problems in loop theory is to determine whether or not all loop isotopes of an arbitrary Osborn loop are themselves Osborn. In [2], Basarab showed that the quotient of a conjugacy closed loop by its nucleus is an abelian group. And in [1], he showed that weak inverse property Osborn loops are characterized by the identity x(yz · x) = ((1/y) · (1/x))\1) · zx. He called these loops generalized Moufang loops.
In [5], A.M. Cheban (Qeban) investigated the structure of two varieties of loops: those loops that satisfy the identity x(xy·z) = yx·xz and those loops that satisfy the identity x(xy · z) = (y · zx)x. We call these the left Cheban identity and the Cheban identity, respectively. Cheban showed that loops satisfying his second identity are generalized Moufang loops and that each of them has an abelian group as an image, which is not surprising in light of Theorem 3.9, below. He also gave an example of a loop satisfying his first identity but that was not Moufang.
As we shall see, Cheban's two varieties have many other strong structural properties; they are also intimately related to conjugacy closed loops.
Our investigations were aided by the automated reasoning tool Prover9 and the finite model builder Mace4 [13]. We have translated most of these computer generated proofs into more "human friendly" form and included them in this paper.
A few of the proofs, though, are quite long, and we have left them untranslated. However, each of these is posted on the first author's website [18], and is clearly referenced in the proofs in this paper. We note that it is common practice to publish complicated untranslated Prover9 proofs [14,19]. This is mathematically sound since the program can be made to output a simple proof object, which can be independently verified by a short lisp program.

Definitions
A loop (Q, ·) is a set Q with a binary operation · such that (i) for each x ∈ Q, the left translation L(x) : Q → Q; y → xy and the right translation R(x) : Q → Q; y → yx are bijections, and (ii) there exists 1 ∈ Q satisfying 1 · x = x · 1 = x for all x ∈ Q. Standard references for the theory of loops are [3,4,16].
A Moufang loop is a loop that satisfies xy · zx = (x · yz)x. A flexible loop satisfies x · yx = xy · x. The left alternative property, denoted by LAP, is given by x · xy = xx · y. The RAP is the mirror identity of the LAP.
The left nucleus of a loop Q is given by N λ (Q) = {a : a · xy = ax · y, ∀x, y ∈ L}. The middle nucleus, N µ (Q), and the right nucleus, N ρ (Q), are defined analogously. The nucleus, then, is given by A loop, Q, is left conjugacy closed, denoted by LCC, if its left translations are closed under conjugation by left translations, i.e., if L(x) −1 L(y)L(x) is itself a left translation for each x, y ∈ Q. This can be expressed equationally as z · yx = ((zy)/z) · zx. Right conjugacy closed, denoted by RCC, is the mirror identity. A loop Q is conjugacy closed, denoted CC, if it is both LCC and RCC. The concept of conjugacy closedness was introduced first by Soikis [21] and later independently by Goodaire and Robinson [9]. In the intervening years a great deal has been discovered about their structural properties.
An especially prominent role in the analysis of CC-loops is assumed by the weak inverse property elements, or WIP elements; i.e., those elements, c such that for every x in the loop we have c(xc) ρ = x ρ , where ρ is the unary operation that gives the right inverse of each element y in the loop, i.e., yy ρ = 1. A loop is power associative if subloops generated by singletons are, in fact, groups. Power associative conjugacy closed loops have especially strong structural properties [10]; this variety is denoted by PACC.
for every x, y in Q 1 . Note that f is an isomorphism if and only if (f, f, f ) is an isotopism.
Finally, an identity α = β is of Bol-Moufang type if (i) the only operation in α, β is ·, (ii) the same 3 variables appear on both sides, in the same order, (iii) one of the variables appears twice on both sides, (iv) the remaining two variables appear once on both sides. For instance, the Moufang law given above, xy · zx = (x · yz)x, is an identity of Bol-Moufang type. The varieties of loops classified by a single identity of Bol-Moufang type were classified in [20]. The Cheban identities are not identities of Bol-Moufang type, since the variables do not appear in the same order on both sides of the equal sign. But they do satisfy the other conditions of the definition. The varieties of loops classified by a single identity of this generalized Bol-Moufang type are classified in [6].
Remark 3.2. The variety of WIP PACC-loops can be axiomatized, in the variety of loops, by the following two identities: (xy · x) · xz = x · (yx · x)z and zx · (x · yx) = (z(x · xy)) · x [17]. Left Cheban loops satisfy the first of these two identities, as the reader may check. Let a be a left nuclear element. Then x · ay = (x/a)a · ay = a((a · x/a)y) = (a(a · x/a))y = ((x/a · a)a)y = xa · y. That is, a is a middle nuclear element. The converse is left to the reader. Now, let b be a commutant element. We have Thus, b is a right nuclear element. For the proof that b is also middle, and hence left, nuclear, see [18]. Finally, let a be a left nuclear element. Then, since R(a) 2 = L(a) 2 , we have x · aa = xa · a = a · ax = aa · x. That is, a 2 is a commutant element. Proof. [18].

Lemma 3.6. A loop is Cheban if and only if it is both left and right Cheban.
By the left Cheban law and the right Cheban law, respectively, we have x(xy·z) = yx · xz = (y · zx)x. For the converse, see [18].
for all x ∈ Q, then Q is, in fact, a Cheban loop.
In preparation for the next theorem, recall that an extra loop is a loop that satisfies the identity x(y · zx) = (xy · z)x. Extra loops are conjugacy closed; in extra loops, squares are nuclear [11]. Proof. [18]. Lemma 3.9. In a cancellative CC-groupoid, the following two conditions are equivalent: The straightforward proof is left to the reader. Proof. Combine Theorem 3.1, Lemma 3.6, and Lemma 3.8. Proof. That Cheban loops are WIP PACC-loops is straightforward and left to the reader. By Basarab's theorem, in CC loops the commutant is contained in the center. And by Theorem 3.10, squares are are contained in the commutant, hence they are also contained in the center. So the factor of a Cheban loop by its center has exponent 2 and is, hence, an abelian group, which finishes the proof of the theorem.
Remark 3.12. A WIP PACC-loop of nilpotency class 2 need not be Cheban, as evidenced by any nonabelian group of odd order and nilpotency class 2 (since in this case, R(x 2 ) = L(x 2 ) will not hold for all x).
Remark 3.13. If L is a left Cheban, right Cheban, or Cheban loop in which every element is either a square or an involution, then L is an abelian group, as the reader may easily check.