scholarly journals Fuzzy Results for Finitely Supported Structures

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1651
Author(s):  
Andrei Alexandru ◽  
Gabriel Ciobanu
Keyword(s):  
Fixed Point ◽  
Fuzzy Sets ◽  
Invariant Set ◽  
Complete Lattices ◽  
Large Sets ◽  
Degree Of Membership ◽  
Function Modelling ◽  
Invariant Group ◽  
Fuzzy Subgroups ◽  
Membership Association

We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function modelling a degree of membership for each element in the crisp set. We define and study the notions of invariant set, invariant complete lattices, invariant monoids and invariant strong inductive sets. The finitely supported (fuzzy) subgroups of an invariant group, as well as the L-fuzzy sets on an invariant set (with L being an invariant complete lattice) form invariant complete lattices. We present some fixed point results (particularly some extensions of the classical Tarski theorem, Bourbaki–Witt theorem or Tarski–Kantorovitch theorem) for finitely supported self-functions defined on invariant complete lattices and on invariant strong inductive sets; these results also provide new finiteness properties of infinite fuzzy sets. We show that apparently, large sets do not contain uniformly supported, infinite subsets, and so they are invariant strong inductive sets satisfying finiteness and fixed-point properties.

Image development in the framework of ξ –complex fuzzy morphisms

2021 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Unit Disk ◽  
Homomorphic Image ◽  
Significant Role ◽  
Original Form ◽  
Mathematical Tool ◽  
Fuzzy Subgroups ◽  
Fundamental Theorems

The study of complex fuzzy sets defined over the meet operator (ξ –CFS) is a useful mathematical tool in which range of degrees is extended from [0, 1] to complex plane with unit disk. These particular complex fuzzy sets plays a significant role in solving various decision making problems as these particular sets are powerful extensions of classical fuzzy sets. In this paper, we define ξ –CFS and propose the notion of complex fuzzy subgroups defined over ξ –CFS (ξ –CFSG) along with their various fundamental algebraic characteristics. We extend the study of this idea by defining the concepts of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –complex fuzzy subgroups and establish fundamental theorems of ξ –complex fuzzy morphisms. In addition, we effectively apply the idea of ξ –complex fuzzy homomorphism to refine the corrupted homomorphic image by eliminating its distortions in order to obtain its original form. Moreover, to view the true advantage of ξ –complex fuzzy homomorphism, we present a comparative analysis with the existing knowledge of complex fuzzy homomorphism which enables us to choose this particular approach to solve many decision-making problems.

scholarly journals Location Selection for Smog Towers Using Zadeh’s Z-Numbers Integrated with WASPAS

2021 ◽  
Author(s):  
Janani Bharatraj
Keyword(s):  
Air Pollution ◽  
Fuzzy Sets ◽  
Degrees Of Freedom ◽  
Assessment Method ◽  
Weighted Sum ◽  
Membership Degree ◽  
Residential Areas ◽  
Location Selection ◽  
Degree Of Membership ◽  
Product Assessment

Fuzzy sets have been extensively researched and results have been developed based on the extensions of fuzzy sets. In this chapter, fuzzy sets and its extensions are discussed. Z-numbers along with weighted sum product assessment method is used to obtain a feasible solution to the location selection problem for installation of smog towers in a densely populated locality. The degrees of freedom namely degree of membership, degree of non-membership and the degree of hesitancy have been expressed as Zadeh’s Z-number with probability quotient for the degrees. Further, ranking of the alternatives based on Z-numbers and WASPAS to allocate smog towers to residential areas stricken by air pollution.

Equational properties of fixed-point operations in cartesian categories: An overview

2019 ◽  
Vol 29 (06) ◽  
pp. 909-925
Author(s):  
Z Ésik
Keyword(s):  
Fixed Point ◽  
Equational Theory ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Finitely Based ◽  
Additive Structure ◽  
Complete Lattices ◽  
Finite Base ◽  
Language Equivalence ◽  
Context Free

AbstractSeveral fixed-point models share the equational properties of iteration theories, or iteration categories, which are cartesian categories equipped with a fixed point or dagger operation subject to certain axioms. After discussing some of the basic models, we provide equational bases for iteration categories and offer an analysis of the axioms. Although iteration categories have no finite base for their identities, there exist finitely based implicational theories that capture their equational theory. We exhibit several such systems. Then we enrich iteration categories with an additive structure and exhibit interesting cases where the interaction between the iteration category structure and the additive structure can be captured by a finite number of identities. This includes the iteration category of monotonic or continuous functions over complete lattices equipped with the least fixed-point operation and the binary supremum operation as addition, the categories of simulation, bisimulation, or language equivalence classes of processes, context-free languages, and others. Finally, we exhibit a finite equational system involving residuals, which is sound and complete for monotonic or continuous functions over complete lattices in the sense that it proves all of their identities involving the operations and constants of cartesian categories, the least fixed-point operation and binary supremum, but not involving residuals.

scholarly journals Robust Invariant Set Analysis of Boolean Networks

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Bowen Li ◽  
Jungang Lou ◽  
Yang Liu ◽  
Zhen Wang
Keyword(s):  
Fixed Point ◽  
Output Feedback ◽  
Invariant Set ◽  
Boolean Networks ◽  
Sufficient Condition ◽  
Lac Operon ◽  
Feedback Controllers ◽  
Control Networks ◽  
E Coli ◽  
Necessary And Sufficient

In this paper, the robust invariant set (RIS) of Boolean (control) networks with disturbances is investigated. First, for a given fixed point, consider a special set called immediate neighborhoods of the fixed point; then a discrete derivative of Boolean functions at the fixed point is used to analyze the robust invariance, based on which a sufficient condition is obtained. Second, for more general sets, the robust output control invariant set (ROCIS) of Boolean control networks (BCNs) is investigated by semitensor product (STP) of matrices. Then, under a given output feedback controller, we obtain a necessary and sufficient condition to check whether a given set is robust control invariant set (RCIS). Furthermore, output feedback controllers are designed to make a set to be a RCIS. Finally, the proposed methods are illustrated by a reduced model of the lac operon in E. coli.

scholarly journals Dynamical Properties of Discrete-Time Background Neural Networks with Uniform Firing Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Min Wan ◽  
Jianping Gou ◽  
Desong Wang ◽  
Xiaoming Wang
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Firing Rate ◽  
Lyapunov Exponents ◽  
Discrete Time ◽  
Invariant Set ◽  
Bifurcation And Chaos ◽  
Dynamical Properties ◽  
Background Input

The dynamics of a discrete-time background network with uniform firing rate and background input is investigated. The conditions for stability are firstly derived. An invariant set is then obtained so that the nondivergence of the network can be guaranteed. In the invariant set, it is proved that all trajectories of the network starting from any nonnegative value will converge to a fixed point under some conditions. In addition, bifurcation and chaos are discussed. It is shown that the network can engender bifurcation and chaos with the increase of background input. The computations of Lyapunov exponents confirm the chaotic behaviors.

scholarly journals Proving Fixed Point Theorems Employing Fuzzy $(\sigma,\mathcal{Z})$-Contractive Type Mappings

2021 ◽  
Author(s):  
Hayel N. Saleh ◽  
Mohammad Imdad ◽  
Salvatore Sessa ◽  
Ferdinando Di Martino
Keyword(s):  
Fixed Point ◽  
Fuzzy Sets ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Contractive Mapping ◽  
Uniqueness Theorems ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Contractive Type

In this article, the concept of fuzzy $(\sigma,\mathcal{Z})$-contractive mapping has been introduced in fuzzy metric spaces which is an improvement over the corresponding concept recently introduced by Shukla et al. [Fuzzy Sets and system. 350 (2018) 85--94]. Thereafter, we utilized our newly introduced concept to prove some existence and uniqueness theorems in $\mathcal{M}$-complete fuzzy metric spaces. Our results extend and generalize the corresponding results of Shukla et al.. Moreover, an example is adopted to exhibit the utility of newly obtained results.

scholarly journals On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Hanan Alolaiyan ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Quotient Group ◽  
Physical Situation ◽  
The Novel ◽  
Complex Fuzzy Set ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
Necessary And Sufficient

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ .

scholarly journals FIXED POINT INDEX IN HYPERSPACES: A CONLEY-TYPE INDEX FOR DISCRETE SEMIDYNAMICAL SYSTEMS

2001 ◽  
Vol 64 (1) ◽  
pp. 191-204 ◽  
Author(s):  
F. R. RUIZ DEL PORTAL ◽  
J. M. SALAZAR
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Metric Spaces ◽  
Invariant Set ◽  
Invariant Sets ◽  
Point Index ◽  
Continuous Map ◽  
Isolated Invariant Set ◽  
Natural Way

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U ⊂ X be an open subset and f: U → X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K ⊂ U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

Fuzzy relation-theoretic contraction principle

2020 ◽  
pp. 1-11
Author(s):  
Waleed M. Alfaqih ◽  
Based Ali ◽  
Mohammad Imdad ◽  
Salvatore Sessa
Keyword(s):  
Fixed Point ◽  
Fuzzy Sets ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fuzzy Relation ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Contractive Mappings ◽  
Fuzzy Fixed Point

In this manuscript, we provide a new and novel generalization of the concept of fuzzy contractive mappings due to Gregori and Sapena [Fuzzy Sets and Systems 125 (2002) 245–252] in the setting of relational fuzzy metric spaces. Our findings possibly pave the way for another direction of relation-theoretic as well as fuzzy fixed point theory. We illustrate several examples to show the usefulness of our proven results. Moreover, we define cyclic fuzzy contractive mappings and utilize our main results to prove a fixed point result for such mappings. Finally, we deduce several results including fuzzy metric, order-theoretic and α-admissible results.

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