A study on Weakly Complemented Nearlattice

Abstract

In this thesis study of the nature of the weakly complemented nearlattice 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 referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upperbound property. Cornish and Noor [8] preferred to Hickman [7] 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 satisfying the descending chain condition has the upper bound property, so all finite semilattices are nearlattices. In lattice theory, it is always very difficult to study the non-distributive and non-modular lattices. Gratzer [12] studied the non-distributive lattices by introducing the concept of distributive, standard and neutral elements in lattices. Cornish and Noor [8] extended those concepts for nearlattices to study non-distributive nearlattices. On the other hand, J.C Varlet [33] 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. In this thesis we have extended the concept of weakly complemented nearlattice in terms of homomorphism theorem.