The Wreath Product of Atoms of the Lattice of Semigroup Varieties

A semigroup variety is called a Cross variety if it is finitely based, is generated by a finite semigroup, and has a finite lattice of subvarieties. It is established in which cases the wreath product of two semigroup varieties each of which is an atom of the lattice of semigroup varieties is a Cross variety. Furthermore, for all the pairs of atoms U and V for which this is possible, either a finite basis of identities for the wreath product UwV is given explicitly, a finite semigroup generating this variety is found and the lattice of subvarieties is described, or it is proved that such a finite characterization is impossible. 1. Statement of the problem and formulation of the main results This paper is devoted to a systematic study of the wreath products of atoms of the lattice of semigroup varieties. We consider three classical conditions for varieties to be finite: having a finite basis of identities, being generated by a finite semigroup, and having a finite lattice of subvarieties. For all the pairs of atoms U and V for which this is possible, a finite basis of identities for the wreath productUwV is given explicitly, a finite semigroup generating this variety is found and the lattice of subvarieties is described. Definition 1.1. A semigroup variety is called a Cross variety if it is finitely based, is generated by a finite semigroup and has a finite lattice of subvarieties. The atoms of the lattice L of all semigroup varieties are well known [16]. These are precisely the varietiesN2 of all semigroups with zero multiplication, Sl of all semilattices, L1 of all semigroups of left zeros, R1 of all semigroups of right zeros andAp of all Abelian groups of prime exponent p. The main result of the paper can be stated as follows. Theorem 1.2. If U and V are atoms of the lattice of semigroup varieties, then the wreath product UwV is a Cross variety, except in the following cases: 1) U = V = Ap; here the variety ApwAp = Ap is finitely based but is not generated by a finite semigroup and has an infinite lattice of subvarieties; 2) U = V = Sl and U = Sl,V = R1; here each of the varieties SlwSl = Sl and SlwR1 is finitely based, is generated by a finite semigroup and has an infinite lattice of subvarieties; 3) U = Sl,V = Ap; here the variety SlwAp is essentially infinitely based, is generated by a finite semigroup and has an infinite lattice of subvarieties. As a by-product, it becomes possible to estimate how big the difference is between the monoid wreath product and the lattice join of two semigroup varieties in the case where the varieties involved in the wreath product are atoms in the lattice of semigroup 2000 Mathematics Subject Classification. Primary 20M07; Secondary 20E22.