Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

Abstract

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras phi:X -> Y, we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping clY:I(Y)-> I(Y), we define a map clY <- n I(X) by A?phi-1(phi(A)clY). We prove that, if "clY" is a weak closure operation (respectively, semi-prime and meet) on I(Y), then so is "clY <-" on I(X). In addition, for mapping clX:I(X)-> I(X), we define a map clX -> on I(Y) as follows: B?phi(phi-1(B)clX). We show that, if "clX" is a weak closure operation (respectively, semi-prime and meet) on I(X), then so is "clX ->" on I(Y).