A certain class of identities on semigroups
J. L. Chrislock · Proceedings of the American Mathematical Society · 1969
We consider here semigroups that satisfy an identityf(x1, , xm) =g(xi, * * *, xs) where the set of variables used in f is not equal to the set of variables used in g. Our main result states the equivalence of the following conditions on a semigroup S: (1) S satisfies an identity of the above form, (2) S satisfies an identity of the form (ZtwtZt)t = Zt, (3) there exists an r > 0 such that S is an ideal extension [2, p. 1371 of a completely simple semigroup [2, Theorem 3.5 and p. 88] whose structure group satisfies xr = 1 by a semigroup that satisfies yr = 0. In light of Clifford's determination of all ideal extensions of a completely simple semigroup [1], the last characterization would, in a sense, be complete if all semigroups with zero which satisfy a relation x8 =0 were known. We do not attempt this here. In what follows we reserve the letters v, w, x, y, z (with or without subscripts) for distinct variables and the letters a, b, c, d (with or without subscripts) for specific members of a semigroup. If m and n are positive integers and a is a function from { 1, * * *, n I to { 1, * * *, m }, then h(xi, * * *, X)=) Xa(l) . . . Xa(n) is a typical semigroup product in the variables x1, X,m. The length n of h will be denoted by | hI, and the set { Xa(l), , Xa(n) } will be written Var (h). If a1, * * , am are elements of a semigroup S, hi(al, * * *, am) denotes that element of S determined by substituting ai for xi (i = 1, * * *, m) in h. All remaining concepts can be found in [2 ].