L-fuzzy Prime and Maximal Congruences in Almost Distributive Lattices

Abstract

In this paper, we define and describe the concept of L-fuzzy prime and maximal congruences in an Almost Distributive lattice (ADL) and discuss its characteristics. Mainly, we establish a one-to-one correspondence between prime (maximal) L-fuzzy congruences of an ADL and the pairs (P, α), where P is a prime (maximal) congruence of an ADL and α is a prime element (dual atom) in a frame, yields the prime (maximal) L-fuzzy congruences of all given ADL. Furthermore, we examine the relationship between prime (maximal) L-fuzzy congruence and L-fuzzy prime (maximal) congruence on an ADL, proving through counter-examples that the converse is untrue.