Study on Standard Ideal and Homomorphism Theorem of a Nearlattice

Abstract

In this thesis the standard ideal of a 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 [4] referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upperbound property. Cornish and Noor [5] preferred to call these semilattices as nearlattices, as the behavior 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. Geatzer [12] studied the non-distributive lattice by introducing the concept of distributive, standard and neutral elements in lattices. Cornish and Noor [5] extended those concepts for nearlattices to study non-distributive nearlattices. This thesis extend the concept of standard ideal of a nearlattices. We also extend the homomorphism theorem of lattices to nearlattices. Finally we generalize two isomorphism theorems of Gratzer, G. and Schmidt, E. T [14] to nearlattices.