The intuitionistic fuzzification in −algebras about the concepts of ideals and subalgebras given with several related characterizations is considered. Some new concepts like intuitionistic fuzzy −ideal ( ), intuitionistic fuzzy −subalgebra ( ), −homomorphism, and intuitionistic fuzzy −ideal ( ) are introduced and some of their descriptions are given in this work. Further, we show some applications on the family of all intuitionistic fuzzy −subalgebras (R) in −algebra like the binary relations ≈, ≈ ] and Γ on (R). Also, their equivalence classes are given and studied.

We first introduce some new notions of Baireness in fuzzy soft topological space (FSTS). Next, their characterizations and basic properties are investigated in this work. The notions of fuzzy soft dense, fuzzy soft nowhere dense, fuzzy soft meager, fuzzy soft second category, fuzzy soft residual, fuzzy soft Baire, fuzzy soft −sets, fuzzy soft −sets, fuzzy soft −nowhere dense, fuzzy soft −meager, fuzzy soft −residual, fuzzy soft −Baire, fuzzy soft −second category, fuzzy soft −residual, fuzzy, fuzzy soft submaximal space, fuzzy soft −space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft −embedded, fuzzy soft −Baire, fuzzy soft almost −spaces, fuzzy soft Borel, and fuzzy soft −algebra are introduced. Furthermore, several examples are shown as well.

Some of the novel algebraic topics explored in this paper, like Permutation Q - algebra, permutation G - part, permutation p-radical, permutation p - semisimple and permutation ideal are discussed and looked into. We show that if ( X, #, {1}) is a permutation Q - algebra, then (λiβ # (λiβ # λjβ)) # λjβ = {1}, ∀λiβ, λjβ ∈ X. Also, in the permutation G - part G(X) of X, the left cancellation law is hold and for any β - set λiβ in Permutation Q - algebra (X, #, {1}), we consider that λiβ belongs to G(X). Additionally permutation implicative, homomorphism, kernel and image of permutation Q - algebras were defined with specific results relating to our unique notions have been developed and examined.

In neutrosophic topological space , The neutrosophic setting for regular generalized is shown as a novel kind neutrosophic set Moreover, Several new ideas of are given and discussed, like neutrosophic closed , neutrosophic set, neutrosophic set, neutrosophic , semi set, neutrosophic , neutrosophic semi . in neutrosophic topological spaces. Moreover, the relationships between the and these new classes in are explained and studied their characters.

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