This work is towards the study of the relationship between fuzzy preordered sets and Alexandrov (left/right) fuzzy topologies based on generalized residuated lattices here the fuzzy sets are equipped with generalized residuated lattice in which the commutative property doesn’t' t hold. Further, the obtained results are used in the study of fuzzy automata theory.

In the current paper, consider the fuzzy normed linear space (X, N) which is defined by Bag and Samanta. First, we construct a new fuzzy topology on this space and show that these spaces are Hausdorff locally convex fuzzy topological vector space. Some necessary and sufficient conditions are established to illustrate that the presented fuzzy topology is equivalent to two previously studied fuzzy topologies.

The aim of this paper is to extend the truth value table of lattice-valued convergence spaces to a more general case and then to use it to introduce and study the quantale-valued fuzzy Scott topology in fuzzy domain theory. Let (L; ∗ ; ") be a commutative unital quantale and let ⊗ be a binary operation on L which is distributive over nonempty subsets. The quadruple (L; ∗ ; ⊗ ; ") is called a generalized GL-monoid if (L; ∗ ; ") is a commutative unital quantale and the operation ∗ is ⊗-semi-distributive. For generalized GL-monoid L as the truth value table, we systematically propose the stratified L-generalized convergence spaces based on stratified L-filters, which makes various existing lattice-valued convergence spaces as special cases. For L being a commutative unital quantale, we define a fuzzy Scott convergence structure on L-fuzzy dcpos and use it to induce a stratified L-topology. This is the inducing way to the definition of quantale-valued fuzzy Scott topology, which seems an appropriate way by some results.

This paper, considers the fuzzy topological subsets, fuzzy topological spaces and introduces a novel concept of fuzzy Hausdorff space and fuzzy manifold space in this regards. Based on these concepts, we present a concept of fuzzy metric Hausdorff spaces and fuzzy metric manifold spaces via the notations of KM-fuzzy metric spaces. This study, generalises the concept of fuzzy metric space to union and product of fuzzy metric spaces in classical logic and in this regard investigates the some product of fuzzy metric fuzzy manifold spaces. We apply the notation of valued-level subsets and make a relation between of topological space, Husdorff space, manifold space and fuzzy topological space, fuzzy Husdorff space and fuzzy manifold space. In final, we extended the fuzzy topological space, fuzzy Husdorff space and fuzzy manifold space to fuzzy metric topological space, fuzzy metric Husdorff space and fuzzy metric manifold space Indeed, this study analyses the notation of fuzzy metric manifold based on valued-level subset.

Small Polygroups are multi-valued systems that satisfy group like axioms. Using the notion of “belonging (Є)” and “quasi-coincidence (q)” of fuzzy points with fuzzy sets, the concept of (Є, Є Vq)-fuzzy subpolygroup is introduced. The study of (Є, Є Vq)-fuzzy normal subpolygroups of a polygroup are dealt with. Characterization and some of the fundamental properties of such fuzzy subpolygroups are obtained. (Є, Є Vq)-fuzzy cosets determined by (Є, Є Vq)-fuzzy subpolygroups are discussed. Finally, we give the definition of a fuzzy subpolygroup with thresholds which is a generalization of an ordinary fuzzy subpolygroup and an (Є, Є, Vq)-fuzzy subpolygroup and discus relations between two fuzzy subpolygroups.

The notion of fuzzy convergence, a prominent tool in fuzzy analysis, has wide applications in fuzzy inference. The present paper examines the distinctive features of the cluster points of fuzzy sequences in various fuzzy topological spaces. A characterization of the fuzzy indiscrete space regarding the cluster points is obtained. Moreover, a detailed study of the cluster points of fuzzy sequences in fuzzy indiscrete, fuzzy discrete, fuzzy co-countable and fuzzy co-finite spaces is presented and characterizations of the cluster points in each space are established. Furthermore, the concepts of maximal cluster points, $c$-sets, $c_m$-sets and fuzzy $c$-sets of fuzzy sequences are proposed and their properties are investigated.

In this paper the concepts of fuzzifying b- irresolute functions and fuzzifying b- compact spaces are characterized in terms of fuzzifying b- open sets and some of their properties are discussed.