Solid varieties of semigroups

Abstract

An identityt≈t′ of terms of a given type is called a hyperidentity of a varietyV (not necessarily of the same type) if whenever the operation symbols occurring int andt′ are replaced by any terms ofV of the appropriate arity, the identity which results holds inV. A varietyV of type ϑ is called solid if every identity ofV also holds as a type ϑ hyperidentity inV. Denecke, Lau, Poschel and Schweigert have shown that a varietyV is solid if and only ifV is a hyperequational class. We use this result, along with the equational description by Denecke and Koppitz of the hyperassociative variety of semigroups, to characterize solid semigroup varieties, and to produce some new examples of such varieties, including some infinite ascending chains in the lattice of solid semigroup varieties.