A study on 0-distributive nearlattice

Abstract

In this thesis study of the nature of the 0-distributive nearlattices is presented. By a nearlattice S we will always mean a meet semilattice together with the property that any two elements possessing a common upper bound, have a supremum. Cornish and Hickman [14] referred this property as the upper bound property and a semilattice of this nature as a semilattice with the upperbound property. Cornish and Noor [15] preferred to call these semilattices as nearlattices, as the behaviour of such a semilattice is close to that of a lattice than an ordinary semilattice. Of course a nearlattice with a largest element is a lattice. Since any semilattice satisf'ing the descending chain condition has the upper bound property, so all finite sernilattices are nearlattices. In lattice theory, it is always very difficult to study the non-distributive and non-modular lattices. Gratzer [20] studied the non-distributive lattices by introducing the concept of distributive, standard and neutral elements in lattices. Cornish and Noor [15] extended those concepts for nearlattices to study non-distributive nearlattices. On the other hand, J.0 Varlet [66] studied another class of non-distributive lattices with 0 by introducing the concept of 0-distributivity. In fact this concept also generalizes the idea of pseudocomplement in a general lattice. This thesis extend the concept of 0-distributivity in a nearlattice to study a larger class of non-distributive nearlattices. A nearlattice S with 0 is called 0-distributive if for all x,y,z ε S with x˄y=0=X˄Z and y˅z; exists imply x˄ (y˅z) = 0. Chapter 1 gives a detailed description of nearlattices. Here we discuss ideals, congruenees, SemiBoolean algebra and many other results on nearlattice which are basic to this thesis. In Chapter 2 we introduce the concept of modular element in a nearlattice. Gratzer and Schmidt [23] introduced the notion of some special elements, e.g. distributive, standard and neutral elements, to study a larger class of non-distributive lattices. Then Cornish and Noor [15] used these concepts to nearlattices. Again Talukder and Noor [64] introduced the notion of modular elements in a join semilattice directed below. The notion of modular element is also applicable for general lattices. In this chapter, we have introduced the concept of modular and strongly distributive elements for nearlattices. i-Iere we have given several characterizations of modular and strongly distributive elements. By studying these elements and ideals we obtained many information on a class of non-distributive nearlattices. Chapter 3 and 4 are the key chapters of this thesis. In Chapter 3 we introduce the 0-distributivity in a nearlattice with 0. We include several characterizations of distributive nearlattices. We prove that a nearlattice S with 0 is 0-distributive if and only if all maximal filter of S are prime. We also show that S is 0-distributive if and only if i(s),the lattice of all ideals of S is pseudocomplemented. Then we include some prime separation properties. In this chapter we also include the notion of semi-prime ideals by extending the notion of 0-distributivity. In lattices, the notion of semi-prime ideals was given by Y. Ray [52]. By using these semi-prime ideals, we generalize the prime separation theorem of nearlattices in terms of annihilator ideals. Finally, we extend the concept of Glivenko congruence for 0-distributive nearlattices as well as for semi-prime ideals to establish a generalized version of prime separation theorem. In chapter 4 we discuss different properties of 0-distributive nearlattices and included several characterizations of these nearlattices Annulets and u-ideals in a distributive lattice have been studied extensively by Cornish [13]. Recently Ayub Ali, Noor and Islam [4], Noor, Ayub Ali and Islam [41] extended this concept for distributive nearlattices. In this chapter we study the annulets and a-ideals in a 0-distributive nearlattice. We give several characterizations of a -ideals. We also include a prime separation theorem for α -ideals. Finally we show that a 0-distributive nearlattice is quasicomplemented if and only if A0(S) (the dual nearlattice of annulets) is a Boolean subalgebra of A(S), where A(S) is the set of all annihilator ideals of S. Moreover, S is sectionally quasicomplemented if and only if A0(S) is relatively complemented. Chapter 5 brings the notions of 0-modular nearlattices. Ayub Ali, Hafizur Rahman and Noor [5], Jayaram [30], Noor, Ayub Ali and Islam [41] and Varlet [65] have studied different properties of 0-distributivity and 0-modularity in lattices and in semilattices. In this chapter we extend their work and include several characterizations of 0-modular nearlattices. Many mathematician including Cornish [Ii] have studied the normal lattices and p-algebras in presence of distributivity. Recently Nag, Begurn and Talukder[38] studied them in presence of 0-dirtributivity. They have generalized many results of S -algebras and D -algebras. Since the idea of pseudocomplementation is not appropriate for a nearlattice, we study the sectional pseudocomplernentation for a nearlattice. In chapter 6 we extend and generalize some results of Nag, Begum and Talukder [38] on D –algebras and S -algebras. We prove that every [0, x] , x ε S is an S -algebra if and only if it is a D -algebra where the nearlattice S is sectionally p-algebra with the condition that [o, x] for each x ε S is 1-distributive and S is 0-modular. We conclude the thesis by giving a characterization of sectionaly S-algebra whenever [0,x} for each x ε S is 1-distributive.