Local finiteness for Green’s relations in semigroup varieties

Abstract

A semigroup S is called locally finite (respectively, periodic) if each finitely generated (respectively, each monogenic) subsemigroup in S is finite. A semigroup variety is locally finite (respectively, periodic) if so are all its members. Clearly, every locally finite variety is periodic but the converse is not true. The issues related to determining extra properties that distinguish locally finite semigroup varieties amongst periodic ones form a vast research area known as Burnside type problems. The reader can find a brief introduction into the main achievements in this area in [34, Chapter 3]; for the present discussion, it suffices to reproduce here just one powerful result by Sapir (a part of [33, Theorem P]).Recall two notions involved in the formulation of Sapir's result. A variety is said to be of finite axiomatic rank if for some fixed n > 0, it can be given by a set of identities involving at most n letters. (For instance, every variety defined by finitely many identities is of finite axiomatic rank.) A semigroup S is said to be a nilsemigroup if S has an element 0 such that s0 = 0s = 0 for every s ∈ S and a power of each element in S is equal to 0. Proposition 1.1. A periodic semigroup variety V of finite axiomatic rank is locally finite if (and, obviously, only if ) all groups and all nilsemigroups in V are locally finite.It is easy to see that all groups in a periodic semigroup variety V form a semigroup variety themselves and so do all nilsemigroups in V. Thus, Proposition 1.