Generalised interval-valued fuzzy ideals and filters in ordered semigroups

Abstract

As a generalisation of fuzzy set, interval-valued fuzzy set has been formed by extending the grade of fuzzy membership function from a point to an interval. This concept has been applied to several algebraic structures, especially in semigroups and ordered semigroups. The importance of the ideas of “belongs to” (E) and “quasi coincident with” ( q ) relations between a fuzzy point and fuzzy set is evident from the volume of researches conducted during the past two decades relating to these concepts. Some researchers employed these concepts in various algebraic structures like BCI-algebra, MV-algebra, semigroups and ordered semigroups by using interval-valued fuzzy set. However, it appears that ordered semigroups and their classification by the properties of interval-valued. Fuzzy concepts are not yet in the literature. In this research, for the concept of “ k ~, quasi coincident with” ( k q~ ) relation between interval-valued fuzzy point and intervalvalued fuzzy subset is used to define the notions of interval-valued. Fuzzy ideal theory and interval-valued. Fuzzy filters in ordered semigroups. Interval-valued. Fuzzy ideal theory refers to interval-valued. Fuzzy ideals, generalised bi-ideal and bi-ideal. These generalised concepts are then linked with ordinary ideals, generalised bi-ideal, bi-ideal and filters by means of level subsets. Furthermore, characterisations of ordered semigroups are studied by the properties of generalised interval-valued fuzzy ideal theory and generalised intervalvalued fuzzy filters. Finally, the notions of t ~ -implication based interval-valued fuzzy ideals and filters are introduced. Thereafter implication based interval-valued fuzzy ideal theory and filters are linked with interval-valued fuzzy ideal theory and filters by using implication operators.