Fuzzy ideals in ordered semigroups and their generalisations

Abstract

The idea of fuzzy sets has opened a new era of research in the world of contemporary mathematics. The proposed concept of fuzzy sets provided for a renewed approach to model imprecision and uncertainty present in phenomena without sharp boundaries. The fuzzification of algebraic structures, particularly ordered semigroups, playa prominent role in mathematics with diverse applications in many applied branches such as computer arithmetic, control engineering, errorcorrecting codes and formal languages. In this background, many researchers initiated the notion of "quasi coincident with" (q) relation between a fuzzy point and a fuzzy set in ordered semigroups. Later a new generalisation of quasi-coincident with relation symbolised as qk where kЄ [0,1) has been introduced. In this thesis, new concepts including fuzzy ideals, fuzzy interior ideals, fuzzy generalised bi-ideals, fuzzy bi-ideals and fuzzy quasi-ideals of type (Є, Є vqk) of ordered semigroup are introduced. Further, ordinary ideals and (Є, Є vqk)-fuzzy ideals are linked using level subset and characteristic function. The results show that in regular, intra-regular and semisimple ordered semigroups both (Є, Є vqk)-fuzzy ideals and (Є, Є vqk)fuzzy interior ideals coincide. The concept of upper/lower parts of (Є, Є vqk)-fuzzy interior ideals is also introduced and furthermore, semisimple, simple and intraregular ordered semi groups are characterised in terms of this notion. The relation between generalised bi-ideals and (Є, Є vqk)-fuzzy generalised bi-ideals is determined. Furthermore, the conditions for the lower part of (Є, Є vqk ) -fuzzy generalised bi-ideal to be a constant function are provided. The characterisations of ordered semigroups by the properties of semiprime (Є, Є vqk) -fuzzy quasi-ideals are investigated. Finally, the classification of ordered semigroups by (Єy,Єy vqg)- fuzzy interior ideals and (Єy , Єy Vqg) -fuzzy interior ideals are determined comprehensively.