Free algebras in discriminator varieties

Abstract

In this paper we prove a representation theorem for the free algebras of certain discriminator varieties which arise when analyzing Boolean products. This result applies to several varieties such as relatively complemented distributive lattices, (generalized) Boolean algebras, Post algebras, P-algebras, B-algebras and in general every variety of R5 lattices. In Section 1 we introduce notation and terminology. In Section 2 we extend the definition of JCl-spectral variety given in [10] to certain V3 classes and we give a characterization of the quasivarieties of a discriminator variety. In Section 3 we prove a theorem which represents the free algebras of the rig-spectral variety as certain Boolean products. In Section 4 we give some applications of this representation theorem.