Fuzzy Interior Research Articles

An interior penalty function method for solving fuzzy nonlinear programming problems

In this article, we investigate fuzzy interior penalty function method for solving fuzzy nonlinear programming problems (FNLPP) based on a new fuzzy arith-metic, unconstrained optimization, and fuzzy ranking on the parametric form of triangular fuzzy numbers (TFN). The main objective of this paper is to solve constrained fuzzy nonlinear programming problems using interior penalty func-tions (IPF) by converting it into unconstrained optimization problems. We prove an important lemma and a convergence theorem for the interior penalty functions method. Interior penalty function techniques favor sites near the boundary of the feasible region in the interior. We present a numerical example of the suggested method and compare the results to those produced by existing methods.

On C̆ech L-fuzzy Rough Proximity Spaces

The aim of this paper is to study the relationship between the category of C̆ech [Formula: see text]-fuzzy rough proximity spaces and the category of C̆ech [Formula: see text]-fuzzy interior spaces which based on rough set models. In particular, we obtain some interesting adjunctions between the considered categories. Also, we give some examples. Moreover, we introduce [Formula: see text]-fuzzy rough grills and discuss some of their properties.

SUP-Hesitant Fuzzy Interior Ideals in \(\Gamma\)-Semigroups

In this paper, we defined the concept \(\mathcal{SUP}\)-hesitant fuzzy interior ideals in \(\Gamma\)-semigroups, which is generalized of hesitant fuzzy interior ideals in \(\Gamma\)-semigroups. Additionally, we study fundamental properties of \(\mathcal{SUP}\)-hesitant fuzzy interior ideals in \(\Gamma\)-semigroups. Finally, we investigate characterized properties of those.

Categories of L-Fuzzy Internal (Enclosed) Relations

This paper reveals some relationships between [Formula: see text]-fuzzy internal (enclosed) relations and [Formula: see text]-fuzzy interior (closure) spaces in a categorical framework. Also, we prove that there exist many interesting Galois connections between the category of all discrete [Formula: see text]-fuzzy interior (closure) spaces and the category [Formula: see text]-fuzzy internal (enclosed) relations. Moreover, the category [Formula: see text]-fuzzy of internal relations, the category of [Formula: see text]-fuzzy quasi-uniform spaces and the category of [Formula: see text]-fuzzy quasi-proximity spaces are investigated.

Relationship Between L-Fuzzy Quasi-Proximity and L-Fuzzy Closure (Interior) Operators

The main purpose of this paper is to study a new relationship among [Formula: see text]-fuzzy quasi-proximity spaces, [Formula: see text]-fuzzy closure spaces and [Formula: see text]-fuzzy interior spaces and their properties are considered, where [Formula: see text] is a complete residuated lattice. Moreover, the Galois connections between [Formula: see text]-fuzzy quasi-proximity spaces, [Formula: see text]-fuzzy closure spaces and [Formula: see text]-fuzzy interior spaces are investigated. We present an example through fuzzy information system which confirms the feasibility of using the proposed approaches.

Rough bipolar fuzzy ideals in semigroups

AbstractFuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-Rs and complete cng-Rs.

Fuzzy sequential topology

We define the sequential fuzzy closure and sequential fuzzy interior of fuzzy subsets of IX by convergence of sequences of fuzzy points. We characterize the fuzzy sequential topology with the sequential fuzzy closure. Furthermore, we compare this topology with the usual fuzzy topology, and prove some basic properties of these concepts.

Some Studies of Multi-Polar Fuzzy Ideals in LA-Semigroups

This article’s main goal is to investigate the concept of multi-polar fuzzy sets (MPF-sets) in LA-semi-groups, which is an extension of bi-polar fuzzy sets (BPF-sets) in LA-semigroups. The main objective of this research is to extend certain significant BPF-set results to MPF-sets results. This article introduces the concepts of multi-polar fuzzy sub LA-semigroups, multi-polar fuzzy quasi-ideals, multi-polar fuzzy bi-ideals, multi-polar fuzzy generalized bi-ideals, and multi-polar fuzzy interior ideals in LA-semigroups. This article also discusses a number of fundamental aspects of multi-polar fuzzy ideals, and we use these aspects to define regular LA-semigroups.

Consequence operators, interior operators and fuzzy relations

In this article the relationship between fuzzy consequence and fuzzy interior operators with fuzzy relations is analyzed. While the situation has been studied in depth if the fuzzy relation is a ⁎-preorder it is not the case for arbitrary fuzzy relations. If R is an arbitrary fuzzy relation, a fuzzy consequence γR and a fuzzy interior δR operators can be derived from R. Using the Representation Theorem for ⁎-preorders, from the columns and rows of R two ⁎-preorders, Rcol and Rrow, can be generated that coincide with the ⁎-preorders generated by γR and δR respectively. A duality result between them and the ⁎-coherence of γR are also shown.

Topological structures and distance functions in complete co-residuated lattices

Most of the research in fuzzy rough sets and fuzzy topological structures have been studied on the basis of fuzzy partially ordered sets. Instead of fuzzy partially ordered sets, the concept of distance functions in complete co-residuated lattices is introduced. Using distance functions, we define Alexandrov pretopology, Alexandrov precotopology and fuzzy interior (fuzzy closure) operators in complete co-residuated lattices, and we investigate their properties. Moreover, we prove that there exist isomorphic categories and Galois correspondence between topological categories.

Ideal theory of semigroups based on ( 3 , 2 ) -fuzzy sets

In this paper, the notions of \((3,2)\)-fuzzy ideal, \((3,2)\)-fuzzy bi-ideal, \((3,2)\)-fuzzy interior ideal and \((3,2)\)-fuzzy \((1,2)\)-ideal of a semigroup are introduced and investigated their properties. The relation between (\((3,2)\)-fuzzy) ideal, bi-ideal, interior ideal and \((1,2)\)-ideal are given. We characterized \((3,2)\)-fuzzy \((1,2)\)-ideal in terms of \(f^3\)-level \(\alpha\)-cut and \(g^2\)-level \(\alpha\)-cut. A necessary and sufficient condition for a subset of a semigroup to be \((1,2)\)-ideal in terms of \((3,2)\)-fuzzy \((1,2)\)-ideal of a semigroup is given.

INTERIOR IDEAL FUZZY SEMIRING

Semiring is one of the ring extensions, which eliminates the inverse axiom in the first operation. One of the topics on the semiring is the ideal interior. This study introduces the concept of the ideal interior semiring and the ideal interior fuzzy semiring. Further, it examined the properties of the ideal fuzzy semiring interior and the nature of the existence of the ideal interior semiring if the ideal fuzzy interior is given.

A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment

<abstract> <p>Theory of $m$-polar fuzzy set deals with multi-polar information. It is used when data comes from $m$ factors $\left({m \ge 2} \right)$. The primary objective of this work is to explore a generalized form of $m$-polar fuzzy subsemigroups, which is $m$-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of $m$-polar fuzzy ternary subsemigroups, $m$-polar fuzzy ideals, $m$-polar fuzzy generalized bi-ideals, $m$-polar fuzzy bi-ideals, $m$-polar fuzzy quasi-ideals and $m$-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every $m$- polar fuzzy bi-ideal of ternary semigroup is an $m$-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of $m$-polar fuzzy ideals, $m$-polar fuzzy bi-ideals.</p> </abstract>

Regularity and normality of (L,M)-fuzzy topological spaces using residual implication

In this paper, the notions of regularity and normality of (L,M)-fuzzy topological spaces are introduced by using residual implication, where L and M are completely distributive De Morgan algebras. It is shown that (L,M)-fuzzy interior operator and (L,M)-fuzzy closure operator can be used to characterize regularity and normality. The relationships among separation axioms of an (L,M)-fuzzy topological space are discussed. Moreover, it is proved that the four separation axioms are equivalent to one another in an (L,M)-fuzzy metric space.

Regular and Intra-Regular Semigroups in Terms of m-Polar Fuzzy Environment

The central objective of the proposed work in this research is to introduce the innovative concept of an m-polar fuzzy set (m-PFS) in semigroups, that is, the expansion of bipolar fuzzy set (BFS). Our main focus in this study is the generalization of some important results of BFSs to the results of m-PFSs. This paper provides some important results related to m-polar fuzzy subsemigroups (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs), m-polar fuzzy quasi-ideals (m-PFQIs) and m-polar fuzzy interior ideals (m-PFIIs) in semigroups. This research paper shows that every m-PFBI of semigroups is the m-PFGBI of semigroups, but the converse may not be true. Furthermore this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semigroups by the properties of m-PFIs and m-PFBIs.

Roughness of (∈*, ∈* ∨ q*)-Fuzzy Ideals in Semigroup

In this paper, we introduce the notions of intervalvalued rough (∈ *, ∈ * ∨ q*)-fuzzy subsemigroup, intervalvalued rough (∈ *, ∈ * ∨ q*)-fuzzy ideals, intervalvalued rough (∈ *, ∈ * ∨ q*)-fuzzy bi-ideal, intervalvalued rough (∈ *, ∈ * ∨ q*)-fuzzy interior ideal and intervalvalued rough (∈ *, ∈ * ∨ q*)-fuzzy quasi ideal as a further generalization of ideals in semigroups and study the properties of these structures.

Notes on Bipolar Interval Valued Intuitionistic Fuzzy Topological Space

In this paper we have introduced Bipolar Interval Valued Intuitionistic Fuzzy Point, Bipolar Interval Valued Intuitionistic Fuzzy Neighbourhood,Bipolar Interval Valued Intuitionistic Fuzzy Interior and Bipolar Interval Valued Intuitionistic Fuzzy Closurein Bipolar Interval Valued Intuitionistic Fuzzy topological space and have verifiedsome of its properties.

An Over view on intuitionistic fuzzy topological spaces

We present a brief overview some fundamental results on the intuitionistic fuzzy topological spaces, and give some introductory results about fuzzy open set, fuzzy closed set, fuzzy neighborhood, fuzzy interior set, fuzzy continuity, fuzzy compactness and fuzzy connectedness in these spaces.

Topological Structure of Fuzzy PU-new Ideal in PU-algebra

In this manuscript, we consider the fuzzification of the notion PU-new ideal of PU-algebra and define the fuzzy topological terms such that fuzzy topology, <i>τ</i>-open fuzzy set, fuzzy neighborhood, fuzzy interior, sequence of fuzzy sets, fuzzy neighborhood system, fuzzy continuity of a function with respect to PU-new ideal of PU-algebra. We explore the new theorems and related properties of above mention notions with respect to PU-new ideal on PU-algebras. Such that for (Ⱬ,<i> τ</i>) to be a TSFP on Ⱬ and the set Ḟ is a fuzzy in Ⱬ and N<SUB>Ḟ</SUB> be a fuzzy neighborhood system of Ḟ then the finite intersection of elements of ‘N<SUB>Ḟ</SUB>’ is also an element of ‘N<SUB>Ḟ</SUB>’ also any fuzzy set of Ⱬ which contains an element of “N<SUB>Ḟ</SUB>” is also an element of “N<SUB>Ḟ</SUB>”. Furthermore we prove the conditions with respect to fuzzy neighborhood, convergence of a sequence of fuzzy sets, fuzzy interior set of a fuzzy set under which a fuzzy set <I>Ḟ</I> is <i>τ</i>-open. We show that how the function Ψ from (Ⱬ₁,<i>τ</i>) to (Ⱬ₂,ω) is fuzzy continuous. We prove that if Ψ is a fuzzy continuous function then for every fuzzy set <I>Ḟ</I> in Ⱬ₁, inverse of each neighborhood of Ψ(<I>Ḟ</I>) is a neighborhood of a fuzzy set <I>Ḟ</I>.

Fuzzy rough sets with a fuzzy ideal

In this paper, we defined the fuzzy upper, fuzzy lower, and fuzzy boundary sets of a rough fuzzy set λ in a fuzzy approximation space (X,R). Based on λ and R, we introduced the fuzzy ideal approximation interior operator intlambdaR and the fuzzy ideal approximation closure operator text {cl}_{lambda }^{R}. We joined the fuzzy ideal notion with the fuzzy approximation spaces, and then introduced the fuzzy ideal approximation closure and interior operators associated to a rough fuzzy set λ. Fuzzy ideal approximation connectedness and the fuzzy ideal approximation continuity between fuzzy ideal approximation spaces are introduced.