Mal'cev products of varieties of completely regular semigroups

Abstract

Abstract Whilst the Mal'cev product of completely regular varieties need not again be a variety, it is shown that in many important instances a variety is in fact obtained. However, unlike the product of group varieties this product is nonassociative. Two important operators introduced by Reilly are studied in the context of Mal'cev products. These operators are shown to generate from any given variety one of the networks discovered by Pastijn and Trotter, enabling identities to be provided for the varieties in the network. In particular the join O V BG of the varieties of orthogroups and of bands of groups is determined, answering a question of Petrich.