A Study on Distributive Nearlattices

Abstract

This thesis studies the nature of distributive nearlattices. By a nearlattice S we will always mean a (lower) semilattice which has the property that any two elements possessing a common upper bound, have a supremum, Cornish and Hickman in their paper [14], referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upper bound property. Cornish and Noor in [15] preferred to call these semilattices as nearlattices as the behaviour of such a semilattice is closer to that of a lattice than an ordiary semilattice. In this thesis we give several results on nearlattices which certainly extend and generalize many results in lattice theory. In chapter 1 we discuss ideals, congruences and other results which are basic to this thesis. We include some characterizations of distributive and modular nearlattices. We generalize the separation properties given by M. H. Stone for distributive lattices. We also show that the set of prime ideals of a nearlattice S is unordered if and only if S is semiboolean.

Chapter 2 discusses the skeletal congruences of a distributive nearlattice. Skeletal congruences on distributive lattices have been studied extensively by Cornish in [11]. Here we extend several results of Cornish for nearlattices. We also introduce the notion of disjutive nearlattices. A distributive nearlattice S with 0 is called disjunctive if for 0 ≤ a < b there is an element x ϵ S such that x ˄ a = 0 and 0 < x ≤ b. Then we give several characterizations of disjunctive nearlattices and semiboolean algebras using skeletal congruences. Finally we show that a distributive naerlattice is semiboolean if and only if θ -----> ker θ is lattice isomorphism of Sc(S) onto KSc(S) whose inverse is the map J´ ---> θ (J).

In chapter 3, we discuss on normal and n-normal nearlattices. Normal lattices have been studied by several authors including Cornish [8] and Monteiro [34]; while n-normal lattices have been studied by Cornish [9] and Davey [16]. In proving some of the results we have used Principle of Localization, which is an extension of lecture note of Dr. Noor on localization. This technique is very interesting and quite different from those of the previous authors.

Chapter 4 studies the multiplier extension (meet translation) of a distributive nearlattjce. Previously multipliers on semilattices and lattices have been studied by several authors e.g, Szasz and Szendrje [54,55,56] Kolibiar [29],Cornjsh [101 and Niemenen [37] on a latice. In a more recent paper, Noor and Cornjsh in [39] studied them on nearlatticee. Here we extend some of their work. We also give a categorjcai result, where we see that the multiplier extension has a functorial character which is entirely different from that of the Lattice Theory, c.f. Cornish [10, theorem 2.41. In section 2 of this chapter we discuss multipliers on sectionally pseudocomplemented distributive nearlattices which are sectionally in B՛n, -1 ≤ n ≤ Ꞷ and generalize a number of results of [10]. We show that S is sectionally in Bn if and only if M(S), the lattice of multipliers is in Bn. Finally we show that for 1 ≤ n < Ꞷ, above conditions are also equivalent to the condition that S is sectionally pseudocomplemented and for any n+1 minimal prime ideals

P1,P2,. ........... ,Pn+1, P1 V P2 V ........... V Pn+1 = S.