Ideals, Congruences and Derivations in

Abstract

This thesis aims to develop a better understanding of ideals, congruence relations and derivations in distributive lattices. We present the definition of a partially ordered set, a lattice and a distributive lattice, and we furnish the relation between these concepts. We introduce the concept of ideal, filter, prime ideal and maximal ideal in lattices . Also we discuss some results related to ideals and their relationship with distributivity. We introduce the concepts of congruence relations, quotient lattices and kernels. In addition we characterize the distributive lattices by kernels. Furthermore we discuss the notion of derivation in lattices and its properties. We compare between derivations in lattices and lattice homomorphisms. Also we present the concepts of d-ideal, injective ideal and discuss two types of congruences on a distributive lattice with respect to derivations . Finally the Stone's result for ideals of a distributive lattice is extended to the case of injective ideals and d-prime ideals