Algebraic structure representations for lattices

TL;DR

This paper explores how algebraic structures can represent finite lattices, building on Birkhoff and Frink's proof, by identifying minimal functions needed for such representations.

Contribution

It characterizes and counts the functions necessary for algebraic structure representations of specific finite lattice types, refining previous constructive proofs.

Findings

Identifies minimal functions for lattice representations

Provides enumeration methods for these functions

Enhances understanding of algebraic structures for lattices

Abstract

For an arbitrary group, the subgroups form a lattice with order determined by set inclusion. Not every lattice is isomorphic to the subgroup lattice for a group. However, Birkhoff and Frink proved that any compactly generated lattice is isomorphic to a subalgebra lattice for some algebraic structure. An algebraic structure is a set A with operations from A^n to A where n$is a non-negative integer. Although the proof by Birkhoff and Frink is constructive, many of the operations described are not needed for an algebraic structure to represent a given lattice. In this paper we utilize concepts in the proof by Birkhoff and Frink to describe and count functions that are used to create algebraic structure representations for certain finite lattice types.