Semisimplicity and the discriminator in bounded BCK-algebras (Q992009)

The purpose of this paper is to characterize certain types of subclasses of the class bBCK of bounded BCK-algebras. The first main theorem of the article states that a relative subvariety \(V\) of bBCK is semisimple iff it satisfies a certain sequence of arithmetic properties, and, moreover, in this case, \(V\) is a variety. The second main theorem of the article states that, given a relative subvariety \(V\) of bBCK, the following equivalences hold: \(V\) is a discriminator variety iff \(V\) is semisimple and congruence-permutable iff \(V\) is a semisimple relative subvariety of involutive BCK-algebras. The author also extends these two main theorems to strong expansions of bBCK and concludes by an example of an involutive BCK-algebra that is not congruence-permutable.