Solid varieties of arbitrary type

Abstract

A solid variety is an equational class in which every identity holds as a hyperidentity as well, meaning that it is satisfied not just by the fundamental operations but also by all terms of the appropriate arity. For type (2), an infinite number of solid varieties (of semigroups) are known, but for other types very few examples of solid varieties are known. In this paper we present several constructions which produce infinite chains of solid varieties. One construction generalizes the normalization of a variety, and gives a method to produce a chain of solid varieties from any given solid variety of type (n). The second construction generalizes the rectangular nilpotent varieties of type (2) to type (n). Finally, we use identities which are consequences of idempotency to construct an infinite chain of solid varieties of any fixed type.