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The intuitionistic fuzzification in ρ-algebras about the concepts of ideals and subalgebras given with several related characterizations is considered. Some new concepts like intuitionistic fuzzy ρ-ideal (IFρi), intuitionistic fuzzy ρ-subalgebra (IFρs), ρ-homomorphism, and intuitionistic fuzzy ρ¯-ideal (IFρ¯i) are introduced and some of their descriptions are given in this work. Further, we show some applications on the family of all intuitionistic fuzzy ρ-subalgebras IFρS(R) in ρ-algebra like the binary relations ≈μ, ≈ν and Γr on IFρS(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 δ-sets, fuzzy soft λσ-sets, fuzzy soft σ-nowhere dense, fuzzy soft σ-meager, fuzzy soft σ-residual, fuzzy soft σ-Baire, fuzzy soft σ-second category, fuzzy soft σ-residual, fuzzy, fuzzy soft submaximal space, fuzzy soft P-space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft A-embedded, fuzzy soft D-Baire, fuzzy soft almost P-spaces, fuzzy soft Borel, and fuzzy soft σ-algebra are introduced. Furthermore, several examples are shown as well.

In this paper we introduced a new family of L. P. O. ℬ ¯ m ˙ , r ^ which represent a generalization of Bernstein oper. for a continuous function on the interval [0,1] . Our goal is to study the approximation properties of this sequence and study the uniformly convergence of it on some continuous functions on a compact set [0,1]. We find a rate of this convergence using a modulus of continuity. Finally we establish a formula of a Voronoviskaja-type asymptotic formula on ℬ ¯ m ˙ , r ^ .