正定关联BCI—代数

Positive Implicative BCI-algebras

本文是作者［１］和［２］的继续，引入了正定关联ＢＣＩ－代数的概念，并证明了：正定关联ＧＢＣＫ－代数类和Ｐ－半单ＢＣＩ－代数类是正定关联ＢＣＩ－代数类的真子类。

This note is a continuation of [1] and [2]. We introduce the concept ofpositive implicative BCI--algebras: a BCI-algebra X is said to be positive implicative if itsatisfies (x * (x * y)) * (y * x) =x * (x * (y * (y * x))). The following results are proved:The class of positive implicative BCK-algebras and the class of p--semisimple algebras areproper subclasses of the class of positive implicative BCI-algebras; Let X be a BCI-alge-bra. If its BCK--part B(X) is a positive implicativc BCK--algebra and if its pure BCI--partA(X) is a p--semisimple BCI--algebra, then X is a positive implicative BCI--algebra; ABCI--algebra is implicative iff it is both commutative and positive implicative.