Extension and structural development of fuzzy (generalized) bi-ideals in ordered semigroups

초록

The concept of ideals in algebraic structures plays an important role in studying their structure. In this paper, we enrich an algebraic structure, which is a generalization of ordered groups called ordered semigroups, by using the more general form of fuzzy subsemigroups and fuzzy (generalized) bi-ideals. In this aim, the idea of (is an element of, is an element of V (k*, q(k)))-fuzzy subsemigroups in ordered semigroups is firstly defined. Then, we show that every fuzzy subsemigroup is a (is an element of, is an element of V (k*, q(k)))-fuzzy subsemigroup but the converse is not true in general. An equivalent condition for a fuzzy set to be a (is an element of, is an element of V (k*, q(k)))-fuzzy subsemigroup is provided. Additionally, the notions of (is an element of, is an element of V(k*, qk))-fuzzy generalized bi-ideals and (is an element of, is an element of V(k*, q(k)))-fuzzy bi-ideals in ordered semigroups are also defined. Several conditions are given for (E, E V (k*, qk))-fuzzy generalized bi-ideals to be (is an element of, is an element of V (k*, q(k)))-fuzzy bi-ideals. We prove that each fuzzy (generalized) bi-ideal is (is an element of, is an element of V (k*, qk))-fuzzy (generalized) bi-ideal; however, the converse is not true, as demonstrated by an example. Furthermore, we provide an equivalent condition for the (is an element of, is an element of V (k*, q(k)))-fuzzy (generalized) bi-ideal. Moreover, we characterize (is an element of, is an element of V(k*, q(k)))-fuzzy (generalized) bi-ideals using level subsets, (is an element of V(k*, qk))-level subsets, and characteristic functions. Finally, by applying the ideal of the (k*, k)-upper part of fuzzy sets, more comprehensive characterizations of (is an element of, is an element of V (k*, q(k)))-fuzzy (generalized) bi-ideals are studied.

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