An Ideal Hilbert Algebras in Positive Implicative BCK-Algebras

Abstract

The notion of BCK-algebras was formulated first in 1966 by K. Iseki, Japanese, and Mathematician. This notion is originated from two different ways. One of the motivations is based on set theory; another motivation is from classical and non-classical propositional calculi. There are many classes of BCK-algebras, for example, sub algebras, bounded BCK-algebras, positive implicative BCK-algebra, implicative BCK-algebra, commutative BCK-algebra, BCK-algebras with condition (S), Griss (and semi-Brouwerian) algebras, quasicommutative BCK-algebras, direct product of BCK-algebras, and so on. The notion of positive implicative BCK-algebras was introduced by K. Iseki in 1975. The notion of ideal in BCK-algebras was formulated by K. Iseki in 1975. The ideal theory plays a fundamental role for the general development of BCK-algebras. Before this article I give a notion an ideal of Hilbert Algebras in BCK-algebras, as well as some propositions, so, here I will give a notion of an ideal of Hilbert algebras in positive implicative BCK-algebras, and some propositions.