On finite hyperidentity bases for varieties of semigroups

Abstract

For any varietyV of semigroups, we denote byH(V) the collection of all hyperidentities satisfied byV. It is natural to ask whether, for a givenV, H(V) is finitely based. This question has so far been answered, in the negative, for four varieties of semigroups: for the varieties of rectangular bands and of zero semigroups by the author in [8]; for the variety of semilattices by Penner in [5]; and for the varietyS of all semigroups by Bergman in [1]. In this paper, we show how Bergman's proof may in fact be used to deal with a large class of subvarieties ofS, namely all semigroup varieties except those satisfyingx 2 =x 2+m for somem. As a first step in the investigation of these exceptional varieties, we also present some hyperidentities satisfied by the variety B1,1 of bands, and, using the same technique, show thatH(V) is not finitely based for any subvarietyV of B1,1. These proofs all exploit the fact that the particular variety in question has hyperidentities of arbitrarily large arity. We conclude with an example of a variety for which even the collection of hyperidentities containing only one binary operation symbol is not finitely based.