Lattices of Equivalence Relations Closed Under the Operations of Relation Algebras

TL;DR

This paper explores which finite lattices can be represented as lattices of equivalence relations closed under specific first-order formulas, demonstrating that all lattices of the form M_n can be represented in this framework.

Contribution

It introduces a new collection of first-order formulas for representing lattices of equivalence relations and proves that all lattices M_n can be represented using this approach.

Findings

Every lattice M_n can be represented as a lattice of equivalence relations closed under the new formulas.

The new collection of formulas defines a distinct class of representations from previous approaches.

Abstract

One of the longstanding problems in universal algebra is the question of which finite lattices are isomorphic to the congruence lattices of finite algebras. This question can be phrased as which finite lattices can be represented as lattices of equivalence relations on finite sets closed under certain first order formulas. We generalize this question to a different collection of first-order formulas, giving examples to demonstrate that our new question is distinct. We then prove that every lattice $\m M_n$ can be represented in this new way. [This is an extended version of a paper submitted to \emph{Algebra Universalis}.]