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) ;

2) ;

3) $x \ast x = 0$;

4) and imply ;

5) implies ;

6) for all . A partial order can then be defined by putting if and only if . A very useful property is .

A BCK-algebra is commutative if it satisfies the identity (cf. also Commutative ring). In this case, , the greatest lower bound of and under the partial order . The BCK-algebra is bounded if it has a largest element. Denoting this element by , one has , the least upper bound of and . In this case, is a distributive lattice with bounds and . A BCK-algebra is positive implicative if it satisfies the identity . This is equivalent to the identity . is called implicative if it satisfies the identity . 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 such that and if and imply . The ideal is implicative if and imply . It is known that always . 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 is an ideal in a BCK-algebra, one can define a congruence relation in by if and only if and . The set of congruence classes then becomes a BCK-algebra under the operation , with as the constant and as the largest element if there exists a largest element . 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 is an ideal and is a lattice filter such that , then there exists a prime ideal such that and . Here, "prime ideal" simply means that if it contains , then it contains either or .

Some of the homological algebra properties of BCK-algebras are known, see [a2]. There is also a close connection between BCK-algebras and commutative -groups with order units (cf. -group). Recall that an element in the positive cone of a commutative -group is an order unit if for each one has for some integer . Let For , let . Then 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