Semilattices with a Partial Binary Operation

Abstract

This thesis Centre’s on a class of certain pseudo complemented partial lattices. This partial lattice is said to be a JP-semilattice. An algebraic structure S = (S, /\, V), where (S, /\) is a semilattice and V is a partial binary operation on S, is said to be a JP-Semilattice if for all x, y, z E S the following axioms hold. (i) x V x exists and x V x = x; (ii) x Vy exists implies y V x exists and x Vy = y V x; (iii) xVy, yV z and (xvy) V z exists implies xv (y V z) exists and (xVy) V z = XV (y V z); (iv) x Vy exists implies x = x I\ (x Vy); (v) x V (x I\ y) exists and x = x V (x I\ y); (vi) y V z exists implies (x I\ y) V (x I\ z) exists. In Chapter 1 we give a background of JP-semilattices. We prove that the set of all ideals of a JP-semilattice is a lattice. Unfortunately, the description of join of two ideals of a JP-semilattice is not good enough like as ideals of a lattice. We close this chapter by giving a relation between JP-homomorphism and order-preserving map. In Chapter 2 we define modular and distributive JP-semilattices. We show that every distributive JP-semilattice is modular but the converse is not neces­sarily true. We prove that a JP-semilattice is non-modular if and only if it has a sublattice isomorphic to the pentagonal lattice. Here we also study the ideals lattice of a modular (distributive) JP-semilattice. We have given some character­izations of modular and distributive JP-semilattice using the ideals lattice. We also give the Stone's Separation Theorem for distributive JP-semilattices. We also prove that if I is an ideal and F is a filter of a distributive JP-semilattice disjoint from I, then there is a minimal prime ideal containing I and disjoint from F.In Chapter 3 we study the congruences of a JP-semilattice. We describe the smallest and largest JP-congruences containing an ideal as a class. Here we char­acterize a distributive JP-semilattice by JP-homomorphism and JP-congruence. We prove the Homomorphism Theorem for JP-semilattices. We have given a description of the smallest JP-congruence containing a filter as a class. The quo­tient of JP-congruence containing a filter as a class is not necessarily a lattice. We impose a condition on the filter, which we call strong filter, to make the quotient JP-semilattice a lattice. Then we study the quotient lattices. Here we give a representation of the set of the prime ideals of the quotient lattice.