Minimal congruence representations of finite distributive lattices

Abstract

In this thesis we will be concerned with determining the best possible representations of finite distributive lattices as congruence lattices of lattices. We first find lower and upper bounds for a finite algebra given its congruence lattice. Secondly, we use the lower bound to determine the minimal representation of a finite product of finite distributive lattices as a congruence lattice of a lattice. This in turn reduces the problem to finding the minimal representation of a product to the minimal representation of the product's directly indecomposable factors. We will then give constructions of minimal representations of particular kinds of directly indecomposable lattices, namely chains of length n. From this we will be able to determine the size of a minimal representation of a product of chains and also a construction for the representation.