Hyperidentity bases for rectangular bands and other semigroup varieties

Abstract

AbstractHyperidentities were introduced by Taylor, and their properties have been studied by Taylor, Bergman, Penner, Graczynska, Schweigert, Wismath, and others. In particular, Penner has produced bases for the hyperidentities of various types satisfied by the variety of semilattices.In this paper, we look at bases for the hyperidentities satisfied by some other varieties of semigroups. We first investigate solid and hyper-associative varieties of semigroups, and use a result of Graczynska's to form a basis for the hyperidentities of type (2) for varieties with these properties. We then produce bases for the hyperidentities satisfied by the variety R B of rectangular bands, and prove that the collection of all hyperidentities satisfied by R B is not finitely based. Finally, we use the R B results to give hyperidentity bases for some varieties of nilpotent semigroups.