Set-Unions Based on t-Conorms in Linear BCK-Algebras

Abstract

In this report, first of all, the authors revisit the concepts of a t-norm and t-conorm in a bounded partially-ordered set, and newly define a quasi-t-conorm in a lower bounded partially-ordered set. Next, we show that some kinds of BCK-algebras are equipped with a t-conorm (or a quasi-t-conorm), and a t-norm. As the main results of this work, it is clarified that a union operation of multi sets is realized by a quasi-t-conorm in a BCK^S-algebra, and a union/intersection operation of fuzzy sets is realized by a t-conorm/a t-norm in a bounded BCK^S-algebra, respectively. Finally, some concrete examples of set operations for multi sets and fuzzy sets are presented.