n-KERNELS OF SKELETAL CONGRUENCES ON A DISTRIBUTIVE NEARLATTICE

Abstract

In this paper, the author studied the skeletal congruences θ^* of a distributive nearlattice S, where * represents the pseudocomplement. Then the author described θ(I)^*, where θ(I) is the smallest congruence of S containing n-ideal I as a class and showed that I^+ is the n-kernel of θ(I)^*. In this paper, the author established the following fundamental results: When n is an upper element of a distributive nearlattice S, the author has shown that the n-kernels of the skeletal congruences are precisely those n-ideals which are the intersection of relative annihilator ideals and dual relative annihilator ideals whose endpoints are of the form x∨n and x∧n respectively. For a central element n of a distributive nearlattice S, the author proved that P_n (S) is disjunctive if and only if the n-kernel of each skeletal congruence is an annihilator n-ideal. Finally, the author discussed that P_n (S) is semi-Boolean if and only if the map θ→Ker_n θ is a lattice isomorphism of SC(S) onto K_n SC(S) whose inverse is the map I→θ(I) where I is an n-ideal and n is a central element of S.