Conditions for embeddability of semigroups in groups

Abstract

We call the words A i and B i defining words. We assume that all defining words are different. We call words of x 1 . . . . . x n without negative powers (including the null word) positive words. By a cusp with respect to (1) we mean a word X for which, for some U l, U 2, V1, and V 2, the words U1XU 2 and VIXV 2 are defining words, and either U 1 ~ U 2 or V l ;~ V 2. The small cancellation condition Cs(p) for (1) is the condition that no defining word is the product of less than p cusps. Following S.I. Adyan [2, 3], we introduce the notions of left and right graphs for the corepresentation (1) (see also [4]). The vertices of both left and right graphs are the letters of the alphabet x 1, x 2 . . . . . x n. For each i E I, we consider the arc ei (/) of the left graph that connects the initial letters of the words A i and B i. Similarly, the arc ei (r) of the right graph connects the final letters of these words. The small cancellation condition D(q) for (1) is the condition that neither the left nor right graphs has cycles with less than q arcs. We say that a semigroup S satisfies the left (right) cancellation condition if, for any x, y, z E S, xy = xz (yx = zx) implies that y = z. If S satisfies both conditions, we say that S is a semigroup with cancellation. The classes Kpq were studied in [1]. By definition, a semigroup S belongs to the class Kpq if it can be specified by a corepresentation (1) that satisfies the small cancellation conditions Cs(P) and D(q). We now recall some results. Adyan [2, 3] proved that all semigroups of the class K1 ~ = tqq Klq, i.e., semigroups without cycles, are embeddable in groups. In [5] Mal'tsev constructed an example of a semigroup with cancellation that belongs to the class K22 but is not embeddable in a group. An even simpler examples was presented in [2]. Such a semigroup is specified by the corepresentation (a, b, c, d, e Iab = cd, aeb =ced) . One of the fundamental results of [1] is Theorem 4.1, which asserts that semigroups of the classes K42 and K33 are embeddable in groups. On the other hand, for all q > 2, Kashintsev [1] constructed examples of semigroups of the classes K2q that are not embeddable in groups. The following two problems were stated in [1]: 1) Are all semigroups that belong to the class I(32 embeddable in groups? 2) Are there semigroups with cancellation in the classes K2q that are not embeddable in groups for q > 2? We answer both of these questions here. We state our fundamental results as follows.