Expansions of finite algebras and their congruence lattices

Abstract

We present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra $(B_0,\dots )$, let $B_1,B_2,\dots ,B_K$ be sets that either intersect $B_0$ or intersect each other at certain points. We construct an \emph{overalgebra} $(A,F_A)$, by which we mean an expansion of $(B_0,\dots )$ with universe $A=B_0\cup B_1\cup \cdots \cup B_K$, and a certain set $F_A$ of unary operations that includes mappings $e_i$ satisfying $e_i^2=e_i$ and $e_i(A)=B_i$, for $0\le i\le K$. We explore two such constructions and prove results about the shape of the new congruence lattices $Con(A,F_A)$ 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.

arXiv comment22 pages, 10 figures