带导子的伪BCI-代数
Pseudo BCI-algebras with derivations
摘要：
本文在伪BCI代数上定义了两类关联导子，研究了它们的性质，并给出了正则II型关联导子的一个刻画。我们还定义了伪BCI代数$A$的$d$ -不变演绎系统的概念，证明了$d$是II型正则导子当且仅当$A$上的每个演绎系统都是$d$ -不变的.证明了伪BCI代数是$p$ -半单的当且仅当唯一的II型正则导子是单位映射.另一个主要结果是证明了一个$p$ -半单伪BCI代数的所有蕴涵导子的集合关于函数合成构成一个交换幺半群。引入了伪BCI代数上的两类对称导子，并证明了在$p$ -半单伪BCI代数的情形下，II型蕴涵导子和II型对称导子的集合是相等的.
英文摘要：
In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a $d$-invariant deductive system of a pseudo-BCI algebra $A$ proving that $d$ is a regular derivation of type II if and only if every deductive system on $A$ is $d$-invariant. It is proved that a pseudo-BCI algebra is $p$-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a $p$-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of $p$-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.
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