Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations

TL;DR

This paper presents an alternative proof that every finite distributive lattice can be represented as a congruence lattice of certain finite algebras with three commutative binary operations, using elementary tools and lattice theory techniques.

Contribution

It adapts a partial algebra construction to provide a new proof of finite distributive lattice representability, emphasizing inequalities and boolean lattices.

Findings

Finite distributive lattices are representable as congruence lattices of certain finite algebras.

The proof uses elementary tools and avoids group-theoretic or closure methods.

The approach may be generalized to broader classes of lattices.

Abstract

A partial algebra construction of Gr\"atzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that every finite distributive lattice is representable, seen as a special case of the Finite Lattice Representation Problem. The construction of this proof brings together Birkhoff's representation theorem for finite distributive lattices, an emphasis on boolean lattices when representing finite lattices, and a perspective based on inequalities of partially ordered sets. It may be possible to generalize the techniques used in this approach. Other than the aforementioned representation theorem only elementary tools are used for the two theorems of this note. In particular there is no reliance on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice to be representable (see William J. DeMeo), such as the closure method.