Abstract
J. C. Varlet introduced the concept of 1-distributive lattices to generalize the notion of dual pseudo complemented lattices. A lattice L with 1 is called a 1-distributive lattice if for all a,b,c is an element of L, a boolean OR b = 1 = a boolean OR c imply a boolean OR (b boolean AND c) = 1. Of course every distributive lattice with 1 is 1-distributive. Also every dual pseudo complemented lattice is 1-distributive. Recently, Shiuly and Noor extended this concept for directed below join semi lattices. A join semi lattice S is called directed below if for all a,b is an element of S, there exists c is an element of S such that c <= a,b. Again Y. Rav has extended the concept of 1-distributivity by introducing the notion of semi prime filters in a lattice. Recently, Noor and Ayub have studied the semi prime filters in a directed below join semi lattice. In this paper we have included several characterizations and properties of 1-distributive join semi lattices. We proved that for a join sub semi lattice A of S, A(1) = {x is an element of S : x boolean OR a = 1 for some a is an element of A} is a semi prime filter of S if and only if S is 1-distributive. We also showed that a directed below join semi lattice with 1 is 1-distributive if and only if for all a,b is an element of S, (a)(1) boolean AND (b)(1) = (d)(1) for some d is an element of S, d <= a,b. Introducing the notion of alpha-filters and using different equivalent conditions of 1-distributive join semi lattices we have given a 'Separation theorem' for alpha-filters.