Disjunctive Nearlattices and Semi-Boolean Algebras

Abstract

A distributive nearlattice S with 0 is disjunctive if 0  a  b implies the existence of xS such that x  a = 0 and 0  x  b . A nearlattice S with 0 is Semi- Boolean if it is distributive and the interval [0, x] is complemented for each xS . In this paper , we establish the following fundamental results : When S is a distributive nearlattice with a central element n , then P (S) n is disjunctive if and only if each dense n -ideal J is both join and meet-dense which is equivalent to the condition that the n -kernel of each skeletal congruence is an annihilator n -ideal. P (S) n is semi-Boolean if and only if for each n -ideal J ,   (J ) = (J ) when n is a central element of S . When S is a distributive nearlattice with a central element n , P (S) n is semi-Boolean if and only if the map   n Ker is a lattice isomorphism of SC(S) onto K SC(S) n whose inverse is the map J (J ) , J is an n -ideal of S .