BCK-代数的理想(Ⅲ)

Ideals in BCK-algebras (Ⅲ)

一个(2,0)型代数【X;*,O】叫做BCK-代数,如果对任意的x,y,z∈X恒有:BCK-1.(x*y)*(x*z)【z*y;BCK-2.x*(x*y)【y;BCK-3.x【x;BCK-4.0【x;BCK-5.x【y和y【x蕴涵x=y。

This note is a continuation of the author's [4]. We give some important properties or ideals in BCK-algebras. Let<X; *, 0> be a BCK-algebra and I be a subset of X. I is said to be a positive implicative ideal if it satisfies (1). 0∈I and (2). (y * x) * z∈I and x * z∈I imply y * z∈I for all x, y, z∈X. I is said to be a commutative ideal if I satisfies: (1) and (3). (x * y)* z∈I and z∈I imply x * (y * (y * x))∈I for all x, y, z∈X. I is an implicative ideal if I satisfies: (1) and (4). (x * (y * x)) *z∈I and z∈I imply x∈I for all x, y, z∈X. Our main results are the following. Theorem 1. If an ideal I or BCK-algebra X is positive implicative (commutative, implicative), then all ideals including I are also positive implicative (commutative, implicative). Corollory. If zero ideal of BCK-algebra X is positive implicative (commutative, implicative), then all ideals of X are positive implicative (commutative, implicative), and so X is a positive implicative (commutative, implicative) BCK-algebra. Theorem 2. If I is an ideal of BCK-algebra X, then quotient algebra <X/I; *, C_0> is positive implicative (commutative, implicative) iff ideal I is positive implicative (commutative, implicative).