M-solid varieties generated by lattices

Abstract

Following W. Taylor, we define an identity $ \varepsilon $ to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity $ \varepsilon $ is hypersatisfied by a variety V, we shall say that $ \varepsilon $ is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, $ \varepsilon $ is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices.