Thematic Studies On Fuzzy Filters In Equality Algebras

Equality algebras were proposed by S.Jenei as a value algebraic structure of higher-order fuzzy logic derived from.EQ-algebras,which provide a wider space for fuzzy type theory.The filter theory of the logical algebras plays an important role in the studying algebraic structures.From a logical point of view,various filters correspond to various sets of provable formulas.This paper will study the fuzzy filter theory on equality algebras and its application.The main contents are as follows:Firstly,we introduce the notion of fuzzy filters and study relative properties of them in equality algebras.In addition,we give the formula for the generation of fuzzy filters on equality algebras,and we solve an open problem given in Kadji A.,Lele C.,Tonga M..Fuzzy prime and maximal filters of residuated lattices[J].Soft Computing,2017,21:1913-1922],that is,is the set of fuzzy filters on the residuated lattice,where lattice is a completely meet distributive,a completely meet distributive lattice?Secondly,we introduce the notions of several kinds of fuzzy filters in equality algebras,study their relative properties7and discus the relationship between them.Finally,we study the extreme fuzzy filters,explore the relationship between extreme fuzzy filters and fuzzy congruences in equality algebras,and discus the uniform topology induced by extreme fuzzy filters.We get the following results:(1)Let E be a lattice equality algebra,where lattice is a completely meet-distributive.Then(FF(E),?,?)is a completely meet-distributive lattice.(2)Let E be an equality algebra.Then f is a fuzzy implicative filter if and only if f is both a fuzzy positive implicative filter and a fuzzy fantastic filter of E.(3)Let E be an equality algebra.Then there is one-to-one correspondence between EX(E)and FC(E).(4)Let E be an equality algebra.Then T={G?E:((?)x?G)((?)A??)s.t.A[x]?G} is an uniform topology on E.(5)Let (?) be an arbitrary family of extreme fuzzy filters of an equality algebra E.Then the space(E,T(?))is a topological equality algebra.