Expansions of finite algebras and their congruence lattices

Abstract

In this paper, we present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra ${\langle B_0, \ldots \rangle}$, let ${B_1,B_2, \ldots , B_K}$ be sets that either intersect B ₀ or intersect each other at certain points. We construct an overalgebra ${\langle A, FA \rangle}$, by which we mean an expansion of ${\langle B_0, \ldots \rangle}$ with universe ${A = B_0 \cup B_1 \cup \ldots \cup B_K}$, and a certain set F _A of unary operations that includes mappings e _i satisfying ${e^2_i = e_i}$ and e _i (A) =  B _i , for ${0 \leq i \leq K}$. We explore two such constructions and prove results about the shape of the new congruence lattices Con${\langle A, F_A \rangle}$ that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant, contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore.