| A congruence relation of a lattice L is an equivalence relation preserving the lattice operations; the set of all congruence relations form a lattice, Con L. The study of the congruence lattices of lattices is one of the fundamental problems in the theory of lattices. In this thesis, we study the relationship between the lattice and its congruence lattice. In Chapter II, we show that if D is a finite distributive lattice with n dual atoms, then there is a lattice L of length 5n such that Con L is isomorphic to D. This answers a problem raised by E. T. Schmidt. We also prove that the bound is best possible in general. Also, we prove that if L is a sectionally complemented lattice, then the length of L is at least 2... |
