Hypervarieties of a given type

Abstract

In this paper we consider the notions of hyperidentities and hypervarieties of a given type τ, without nullary operations, originated by J. Aczel [1], V. D. Belousov [2], W. D. Neumann [12] and W. Taylor [21]. Solid varieties are defined. Their equations correspond to hyperidentities and they form a complete sublattice of the lattice L(τ) of all varieties of type τ. A completeness theorem for hyperidentities is formulated in analogue to G. Birkhoff [3]. The technique of weak isomorphisms, introduced by E. Marczewski and A. Goetz [7] assists us in partially answering a problem of W. Taylor (Problem 4, [21]).