交换BCK-代数的新型软(素)理想

New Type of Soft(Prime)Ideals in Commutative BCK-Algebras

软集是处理不确定性问题的一个重要的数学工具。将软集的参数集赋予交换BCK-代数(即为交换亚BCI-代数),提出了交换BCK-代数的新型软素理想、软集的零化子、新型对合软理想等新概念。定义了两个新的合成运算,并利用合成运算对交换BCK-代数的新型软理想进行刻画。研究了交换BCK-代数的新型软理想在偏序关系下的若干性质以及软集的零化子和新型对合软理想的性质。举例说明了交换BCK-代数的新型软素理想的存在性,以及它与通常的软素理想的不同。举例说明了一个软集是交换BCK-代数的新型软素理想与它的水平集是素理想不是充要条件,这与通常的模糊代数的结果不同。给出了交换BCK-代数的新型软素理想的一些等价刻画,并讨论了它的同态像与原像的性质。

The soft set theory is an important mathematical tool for dealing with uncertainty.By endowing a par ameter set as a commutative BCK-algebra(that is commutative weak-BCI-algebra),the notions of a new type of soft prime ideals,annihilators of soft sets and new type of involutory soft ideals in commutative BCK-algebras are introduced.Two new compositional operations are defined and used to characterize the new type of soft ideals in commutative BCK-algebras.By using partial ordering on commutative BCK-algebras,some properties of the new type of soft ideals are studied.Properties of annihilators of soft sets and new type of involutory soft ideals are obtained.The existence of a new type of soft prime ideals in commutative BCK-algebras and its difference from the standard soft prime ideals are illustrated with examples.It is shown that a soft set is a new type of soft prime ideals in commutative BCK-algebras and its level set is a prime ideal is not a necessary and sufficient condition,which is different from the results of the usual fuzzy algebra.Some equivalent characterizations of the new type of soft prime ideals in commutative BCK-algebras are given.Furthermore,the properties of its homomorphism image and inverse image are discussed.