BCK-代数的J-扩张

J-Extension of BCK-Algebras

本文给出一种新的BCK-代数的扩张方法,并用这种方法给出122种6阶1型BCK-代数.

Let X=<X;*, 0> be a BCK-algebra of order n(≥3). If there exists a standard sequence of X, say, 0,a1,a2, …,an-1 and a subscript i0 with 0≤i0≤n-2 such that holds for each j with 0≤j≤n-2, then X is called a BCK-algebra with condition (J). We call an-1 a maximal element with condition (J). Our main results are. Theorem 1 Let X=<X;*,0> be a BCK-algebra of order n and with condition (J), as described above. Let u?X. Set X'=X∪{u}. We define an operation *' ?n X' as follows: x*'y=x*y if x, y∈X; u*'x=u if x∈X; x*'u=0 if x∈X-{an-1}; an-1*'u=an-1*ai0 and u*'u=0. Then X'=<X';*',0> is a BCK-algebra of order n+1. We call X' a J-extension of X (with respect to an-1). Theorem 2 If X1 and X2 are BCK-algebras with the condition (J) such that they are not isomorphic, then their J-extensions are also not isomorphic. Let X be a BCK-algebra with condition (J), and let (?) be the set of the maximal elements of X satisfying the condition (J) . Then under the action of the automorphism group of X, (?) is divided into disjoint orbits. The number of these orbits is denoted by c(X). Then we have. Theorem 3 If X is a finite BCK-algebra with the condition (J), then the number of J-extensions of X is exactly equal to c(X) . In this paper we have listed 122 BCK-algebras of order 6 and of type 1, by constructing J-extensions of BCK-algebras of order 5 satisfying the condition (J).