非交换的正关联BCK-代数

On Non-commutative Positive Implicative BCK-Algebras

本文解决了关于非交换的正关联BCK-代数的真类问题、基数问题和真子类问题。作者使用有限BCK-代数的局部分析法,确定了阶不超过5的非交换正关联BCK-代数的全部类型,讨论了C型和非C型的非交换正关联BCK-代数的有关问题。

Let (X,≤,0) be a partially ordered set with a smallest element 0. For define x,y∈X, define Then we get a BCK-algebra in which the original order is consistent with that defined by the multiplication *. This fact was pointed out by W.H. Cornish. So we say that such BCK-algebras are of C-type. In this paper we show that the BCK-algebras of C-type are positiveimplicative. We prove following theorems. Theorem 1. The positive implicative BCK-algebras of C-type are non-commutative except one which is implicative. Theorem 2. For any cardinal number γ,there exists a non-commutative positive implicative BCK-algebra X such that |X|=γ. Theorem 3. Non-commutative positive implicative BCK-algebras do not form a variety. Theorem 4. If X is a non-commutative positive implicative BCK-algebra and is not of C-type, then |X|≥5. Theorem 5. There are exact four non-commutative positive implicative BCK-algebras which have order 5 and are not of C-type. Theorem 6. For any cardinal number γ≥5 there exist at least four non-commutative positive implicative BCK-algebras which have order γ and are not of C-type.