Core Concepts

To understand the FLRP, we need a shared vocabulary. Here are the fundamental definitions from lattice theory and universal algebra.

Finite Lattice

A lattice is a partially ordered set where every two elements have a unique least upper bound (join, written \(x \lor y\)) and a unique greatest lower bound (meet, written \(x \land y\)). A finite lattice is simply a lattice with a finite number of elements. They are the fundamental "shapes" we are trying to represent.

Algebra and Finite Algebra

An algebra \(\mathbf{A}\) consists of a non-empty set \(A\) called the universe and a collection of finitary operations on \(A\). A finite algebra is one whose universe \(A\) is a finite set. Think of groups, rings, and vector spaces as examples of algebras.

Congruence Lattice (\(\text{Con } \mathbf{A}\))

A congruence on an algebra \(\mathbf{A}\) is an equivalence relation on its universe that is compatible with all of its operations. The set of all congruences on \(\mathbf{A}\), ordered by set inclusion, forms a lattice itself—the congruence lattice of \(\mathbf{A}\), denoted \(\text{Con } \mathbf{A}\). This lattice captures the essential "internal symmetries" of the algebra.

Algebraic Lattice

A lattice is algebraic if it is complete and every element is a join of compact elements. A key theorem by Birkhoff and Frink (1948) states that the congruence lattice of any algebra is always an algebraic lattice. Every finite lattice is, by definition, an algebraic lattice.