An Application of Generalized Fuzzy Ideals in Ordered Semigroups

Ordered semigroups are algebraic systems consisting of a nonempty set, an associative binary operation, and a partial order compatible with this binary operation. This concept is a generalization of semigroups. One mathematical tool used to study ordered semigroups is the concept of ideals. It turns out that ordered semigroups can be decomposed based on their regularities using various kinds of ideals. The concept of fuzzy sets is one of the mathematical tools used to investigate ordered semigroups, specifically through so-called fuzzy ideals, which are more appropriate than set-theoretical ideals. It is known that the concepts of $(\alpha, \beta)$-fuzzy $(m, n)$-ideals and $(\alpha, \beta)$-fuzzy $n$-interior ideals are generalizations of various types of ideals and many kinds of fuzzy ideals in ordered semigroups. In this paper, we apply the notions of $(\alpha, \beta)$-fuzzy $(m, n)$-ideals and $(\alpha, \beta)$-fuzzy $n$-interior ideals to classify ordered semigroups into classes based on their regularities, using the meaning of characteristic functions.