# 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.

## 2010 Mathematics Subject Classification

Primary: 08A30 Secondary: 08A60 06B10## Key words and phrases

congruence lattice finite algebra finite lattice representations## Preview

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