Associativity of products of existence varieties of regular semigroups Original Research Article

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 image, the e-variety image o image, where o denotes the Malʹcev product within the class of regular semigroups and image denotes the e-variety of left groups, is actually equal to the e-variety image generated by all wreath products of the form G circle times operator T, where image, the e-variety of all groups, and image. It was also shown that if image denotes the e-variety of left zero semigroups and image the e-variety of all semilattices, then image o image is equal to the e-variety image generated by certain subsemigroups of the wreath products of the form S circle times operator T, where image and image. In this paper, the e-variety generated by the regular parts of the wreath products of the form image, image and image, where image, image and image 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 Malʹcev 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.