Expansions of finite algebras and their congruence lattices

TL;DR

This paper introduces a new method for constructing finite algebras with specific congruence lattices, expanding the understanding of finitely representable lattices and proposing a novel approach to discovering new classes of such lattices.

Contribution

It presents a novel construction of finite algebras called overalgebras and explores their congruence lattices, offering new classes of finitely representable lattices and a new approach to their discovery.

Findings

Constructed overalgebras with prescribed congruence lattices

Identified new classes of finitely representable lattices

Proposed a novel method for discovering representable 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 $0\leq i \leq K$. 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.