BCK-algebra

Algebras originally defined by K. Iséki and S. Tanaka in [a7] to generalize the set difference in set theory, and by Y. Imai and Iséki in [a5] as the algebras of certain propositional calculi. A BCK-algebra may be defined as a non-empty set $X$ with a binary relation $\ast$ and a constant $0$ satisfying the following axioms:

1) $\{(x\ast y) \ast (x\ast z)\} \ast (z \ast y) = 0$;

2) $\{x \ast (x \ast y) \} \ast y = 0$;

3) $x \ast x = 0$;

4) $x \ast y = 0$ and $y \ast x = 0$ imply $x = y$;

5) $x \ast 0 = 0$ implies $x = 0$;

6) $0 \ast x = 0$ for all $x$. A partial order $\le$ can then be defined by putting $x \le y$ if and only if $x \ast y = 0$. A very useful property is $(x \ast y) \ast z = (x \ast z) \ast y$.

A BCK-algebra is commutative if it satisfies the identity $x \ast (x \ast y) = y \ast (y\ast x)$ (cf. also Commutative ring). In this case, $x \ast (x \ast y) = x \wedge y$, the greatest lower bound of $x$ and $y$ under the partial order $\le$. The BCK-algebra is bounded if it has a largest element. Denoting this element by $1$, one has $1 \ast \{(1\ast x) \wedge (1\ast y)\} = x \vee y$, the least upper bound of $x$ and $y$. In this case, $X$ is a distributive lattice with bounds $0$ and $1$. A BCK-algebra is positive implicative if it satisfies the identity $(x\ast y)\ast z = (x\ast z) \ast (y\ast z)$. This is equivalent to the identity $x \ast y = (x\ast y)\ast y$. $X$ is called implicative if it satisfies the identity $x \ast (y \ast x) = x$. Every implicative BCK-algebra is commutative and positive implicative, and a bounded implicative BCK-algebra is a Boolean algebra.

An ideal of a BCK-algebra is a non-empty set $I$ such that $0 \in I$ and if $x \ast y \in I$ and $y \in I$ imply $x \in I$. The ideal is implicative if $(x\ast y) \ast z \in I$ and $y \ast z \in I$ imply $x \ast z \in I$. It is known that always $(x \ast z) \ast z \in I$. Note that in a positive implicative BCK-algebra, every ideal is implicative. Implicative ideals are important because in a bounded commutative BCK-algebra they are precisely the ideals for which the quotient BCK-algebras are Boolean algebras. Here, if $I$ is an ideal in a BCK-algebra, one can define a congruence relation in $X$ by $x \sim y$ if and only if $x \ast y \in I$ and $y \ast x \in I$. The set $X/I$ of congruence classes then becomes a BCK-algebra under the operation $[x]\ast [y] = [x\ast y]$, with $[0]$ as the constant and $[1]$ as the largest element if there exists a largest element $1$. Some, but not all, of the well-known results on distributive lattices and Boolean algebras hold in BCK-algebras, in particular in bounded commutative BCK-algebras. For example, the prime ideal theorem holds for bounded commutative BCK-algebras, that is, if $I$ is an ideal and $F$ is a lattice filter such that $I \cap F = \emptyset$, then there exists a prime ideal $J$ such that $I \subset J$ and $J \cap F = \emptyset$. Here, "prime ideal" simply means that if it contains $x \wedge y$, then it contains either $x$ or $y$.

Some of the homological algebra properties of BCK-algebras are known, see [a2]. There is also a close connection between BCK-algebras and commutative $l$-groups with order units (cf. $l$-group). Recall that an element $u$ in the positive cone $G_+$ of a commutative $l$-group $G$ is an order unit if for each $x \in G$ one has $x \le nu$ for some integer $n$. Let $G(u) = \{x \in G : 0 \le x < u\}$. For $x, y \in G(u)$, let $x \ast y = (x-y)_+ = (x-y) \vee 0$. Then $G(u)$ is a commutative BCK-algebra.

Fuzzy ideals of BCK-algebras are described in [a3] and [a4]. General references for BCK-algebras are [a6] and [a7].

References

[a1]	C.S. Hoo, P.V. Ramana Murty, "The ideals of a bounded commutative BCK-algebra" Math. Japon. , 32 (1987) pp. 723–733
[a2]	C.S. Hoo, "Injectives in the categories of BCK and BCI-algebras" Math. Japon. , 33 (1988) pp. 237–246
[a3]	C.S. Hoo, "Fuzzy ideals of BCI and MV-algebras" Fuzzy Sets and Systems , 62 (1994) pp. 111–114
[a4]	C.S. Hoo, "Fuzzy implicative and Boolean ideals of MV-algebras" Fuzzy Sets and Systems , 66 (1994) pp. 315–327
[a5]	Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 19–22
[a6]	K. Iséki, S. Tanaka, "Ideal theory of BCK-algebras" Math. Japon. , 21 (1976) pp. 351–366
[a7]	K. Iséki, S. Tanaka, "An introduction to the theory of BCK-algebras" Math. Japon. , 23 (1978) pp. 1–26