Hilbert Algebras in Negative Implicative BCK-Algebras

摘要

The notion of BCK-algebras was formulated first in 1966 by 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 nonclassical propositional calculi. There are many classes of BCK-algebras, for example, subalgebras, 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 Iseki in 1975. In previous studies, scholars gave the definition of the positive implicative BCK-algebras, and its characterizations, and the relationship between other BCK-algebra, 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 Hilbert algebras in negative implicative BCK-algebras, and some propositions.