LFuzzy Ideals and LFuzzy Congruence Relations in Universal Algebra

Gezahagne, Mulat

dc.contributor.author	Gezahagne, Mulat
dc.date.accessioned	2020-09-15T12:38:09Z
dc.date.available	2020-09-15T12:38:09Z
dc.date.issued	2020-09-15
dc.identifier.uri	http://hdl.handle.net/123456789/11216
dc.description.abstract	In this thesis, we introduce the notion of fuzzy ideals in a more general context in universal algebras by the use of ideal terms. Fuzzy ideals generated by a fuzzy set are characterized from the fuzzy point of view as well as from the algebraic point of view. It is shown that the class of fuzzy ideals of an algebra A of a given type F forms an algebraic closure fuzzy set system together with the inclusion ordering of fuzzy sets. The commutator of fuzzy ideals in universal algebras is deﬁned as a common abstraction of the product of fuzzy normal subgroups in groups, fuzzy ideals in rings, etc. Using this commutator, we deﬁne and characterize fuzzy prime ideals, fuzzy semiprime ideals, maximal fuzzy ideals, the radical of fuzzy ideals and the space of fuzzy prime ideals in universal algebras. On the other hand, we deal with fuzzy congruence relations and their classes so-called fuzzy congruence classes in universal algebras. We characterize fuzzy congruence relations generated by a fuzzy relation and we give a representation for fuzzy congruence relations using crisp congruences. Mainly, we make a theoretical study on fuzzy congruence classes of algebras in different varieties. Several Mal’cev type characterizations are given for a fuzzy subset of an algebra in a given variety to be a class of some fuzzy congruence. Particularly, ﬁnite characterizations are given for fuzzy congruence classes in regular and permutable varieties. We also introduce the notion of fuzzy cosets in universal algebras by the use of coset terms. It is shown that, fuzzy ideas and more generally fuzzy congruence classes are the natural examples of fuzzy cosets. But the converse does not hold in general. We give sufﬁcient conditions for fuzzy cosets to be a class of some fuzzy congruence relation. Moreover, the theory of fuzzy cosets is applied to characterize permutable varieties.	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics.	en_US
dc.title	LFuzzy Ideals and LFuzzy Congruence Relations in Universal Algebra	en_US
dc.type	Thesis	en_US