Associativity of products of existence varieties of regular semigroups

Abstract

An e-variety is a class of regular semigroups that is closed under the formation of direct products, homomorphic images and regular subsemigroups. In a previous paper, the authors established that, for any nongroup e-variety V , the e-variety LG o V , where o denotes the Mal'cev product within the class of regular semigroups and LG denotes the e-variety of left groups, is actually equal to the e-variety G ⊗ V generated by all wreath products of the form G ⊗ T, where G ϵ G , the e-variety of all groups, and T ϵ V . It was also shown that if LF denotes the e-variety of left zero semigroups and J the e-variety of all semilattices, then LF o V is equal to the e-variety J ⊗ ∗ V generated by certain subsemigroups of the wreath products of the form S ⊗ T, where S ϵ J and T ϵ V . In this paper, the e-variety generated by the regular parts of the wreath products of the form RF ⊗ V , RB ⊗ V and BJ ⊗ V , where RF , RB and BJ denote the e-variety of right zero semigroups, rectangular bands and completely simple semigroups respectively, are studied and are found, in general, to fall far short of the corresponding Mal'cev products. An important tool is the associativity of the wreath product of e-variety under certain conditions and a substantial part of the paper is devoted to this issue.