scholarly journals A characterization of regular, intra-regular, left quasi-regular and semisimple hypersemigroups in terms of fuzzy sets

2017 ◽  
Vol 26 (1) ◽  
pp. 46-56
Author(s):  
Niovi Kehayopulu
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Subset ◽  
Fuzzy Subsets

Abstract We prove that an hypersemigroup H is regular if and only, for any fuzzy subset f of H, we have f ≼ f ο 1 ο f and it is intra-regular if and only if, for any fuzzy subset f of H, we have f ≼1 ο f ο f ο 1. An hypersemigroup H is left (resp. right) quasi-regular if and only if, for any fuzzy subset f of H we have f ≼ 1 ο f ο 1 ο f (resp. f ≼ f ο 1 ο f ο 1) and it is semisimple if and only if, for any fuzzy subset f of H we have f ≼ 1 ο f ο 1 ο f ο 1. The characterization of regular and intra-regular hypersemigroups in terms of fuzzy subsets are very useful for applications.

Normalizing Possibility Distributions Using t-Norms

2003 ◽  
Vol 11 (03) ◽  
pp. 343-360
Author(s):  
J. Recasens ◽  
J. Lawry
Keyword(s):  
Fuzzy Sets ◽  
Normalization Method ◽  
Fuzzy Subset ◽  
View Point ◽  
New Approach ◽  
Possibility Distributions ◽  
Fuzzy Subsets

A new approach to normalizing fuzzy sets is introduced where it is assumed that the normalization method is compatible with a given t-norm. In this context it is proved that the most usual ways to normalize fuzzy subsets correspond to the most common t-norms. For a given fuzzy subset μ, the corresponding normalized fuzzy subset [Formula: see text] can be viewed as the distribution of μ conditioned on the (degree of) existence of its elements with maximal membership. From this view point we investigate the less specific normal fuzzy subset of X among the most similar fuzzy subsets to μ and the normal fuzzy subset generating the same fuzzy T-preorder as μ.

Granular Computing and Human-Centricity in Computational Intelligence

2012 ◽  
pp. 13-27
Author(s):  
Witold Pedrycz
Keyword(s):  
Fuzzy Sets ◽  
Granular Computing ◽  
Knowledge Generation ◽  
Qualitative Assessment ◽  
Information Granules ◽  
Knowledge Based ◽  
Performance Indexes ◽  
Numeric Data ◽  
Type Information

Information granules and ensuing Granular Computing offer interesting opportunities to endow processing with an important facet of human-centricity. This facet implies that the underlying processing supports non-numeric data inherently associated with the variable perception of humans. Systems that commonly become distributed and hierarchical, managing granular information in hierarchical and distributed architectures, is of growing interest, especially when invoking mechanisms of knowledge generation and knowledge sharing. The outstanding feature of human centricity of Granular Computing along with essential fuzzy set-based constructs constitutes the crux of this study. The author elaborates on some new directions of knowledge elicitation and quantification realized in the setting of fuzzy sets. With this regard, the paper concentrates on knowledge-based clustering. It is also emphasized that collaboration and reconciliation of locally available knowledge give rise to the concept of higher type information granules. Other interesting directions enhancing human centricity of computing with fuzzy sets deals with non-numeric semi-qualitative characterization of information granules, as well as inherent evolving capabilities of associated human-centric systems. The author discusses a suite of algorithms facilitating a qualitative assessment of fuzzy sets, formulates a series of associated optimization tasks guided by well-formulated performance indexes, and discusses the underlying essence of resulting solutions.

scholarly journals A p-Ideal in BCI-Algebras Based on Multipolar Intuitionistic Fuzzy Sets

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 993
Author(s):  
Jeong-Gon Lee ◽  
Mohammad Fozouni ◽  
Kul Hur ◽  
Young Bae Jun
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Finite Degree ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Ideal ◽  
The Relationship

In 2020, Kang, Song and Jun introduced the notion of multipolar intuitionistic fuzzy set with finite degree, which is a generalization of intuitionistic fuzzy set, and they applied it to BCK/BCI-algebras. In this paper, we used this notion to study p-ideals of BCI-algebras. The notion of k-polar intuitionistic fuzzy p-ideals in BCI-algebras is introduced, and several properties were investigated. An example to illustrate the k-polar intuitionistic fuzzy p-ideal is given. The relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal is displayed. A k-polar intuitionistic fuzzy p-ideal is found to be k-polar intuitionistic fuzzy ideal, and an example to show that the converse is not true is provided. The notions of p-ideals and k-polar ( ∈ , ∈ ) -fuzzy p-ideal in BCI-algebras are used to study the characterization of k-polar intuitionistic p-ideal. The concept of normal k-polar intuitionistic fuzzy p-ideal is introduced, and its characterization is discussed. The process of eliciting normal k-polar intuitionistic fuzzy p-ideal using k-polar intuitionistic fuzzy p-ideal is provided.

scholarly journals The Interaction between Fuzzy Subsets and Groupoids

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Seung Joon Shin ◽  
Hee Sik Kim ◽  
J. Neggers
Keyword(s):  
Binary System ◽  
Fuzzy Subset ◽  
Linear Combinations ◽  
Real Linear ◽  
Fuzzy Subsets

We discuss properties of a class of real-valued functions on a setX2constructed as finite (real) linear combinations of functions denoted asX,*;μ, whereX,*is a groupoid (binary system) andμis a fuzzy subset ofXand whereX,*;μx,y≔μx*y-minμx,μy. Many properties, for example,μbeing a fuzzy subgroupoid ofX,*, can be restated as some properties ofX,*;μ. Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals ofBinX;□for example.

Granular Computing and Human-Centricity in Computational Intelligence

2012 ◽  
pp. 1721-1735
Author(s):  
Witold Pedrycz
Keyword(s):  
Fuzzy Sets ◽  
Granular Computing ◽  
Knowledge Generation ◽  
Qualitative Assessment ◽  
Information Granules ◽  
Knowledge Based ◽  
Performance Indexes ◽  
Numeric Data ◽  
Type Information

Information granules and ensuing Granular Computing offer interesting opportunities to endow processing with an important facet of human-centricity. This facet implies that the underlying processing supports non-numeric data inherently associated with the variable perception of humans. Systems that commonly become distributed and hierarchical, managing granular information in hierarchical and distributed architectures, is of growing interest, especially when invoking mechanisms of knowledge generation and knowledge sharing. The outstanding feature of human centricity of Granular Computing along with essential fuzzy set-based constructs constitutes the crux of this study. The author elaborates on some new directions of knowledge elicitation and quantification realized in the setting of fuzzy sets. With this regard, the paper concentrates on knowledge-based clustering. It is also emphasized that collaboration and reconciliation of locally available knowledge give rise to the concept of higher type information granules. Other interesting directions enhancing human centricity of computing with fuzzy sets deals with non-numeric semi-qualitative characterization of information granules, as well as inherent evolving capabilities of associated human-centric systems. The author discusses a suite of algorithms facilitating a qualitative assessment of fuzzy sets, formulates a series of associated optimization tasks guided by well-formulated performance indexes, and discusses the underlying essence of resulting solutions.

A Characterization of Optimal Fuzzy Sets in Fuzzy Optimization

1993 ◽  
Vol 1 (4) ◽  
pp. 313-317 ◽  
Author(s):  
Sie-Keng Tan ◽  
Pei-Zhuang Wang
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Optimization

FUZZY SIMILARITY AND MEASURE OF SIMILARITY WITH LUKASIEWICZ IMPLICATOR

2008 ◽  
Vol 04 (02) ◽  
pp. 191-206 ◽  
Author(s):  
ISMAT BEG ◽  
SAMINA ASHRAF
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Fuzzy Measure ◽  
Fuzzy Similarity ◽  
Measure Of Similarity ◽  
Universe Of Discourse ◽  
Fuzzy Subsets

Given a universe of discourse X. A fuzzy similarity mapping ST,Inc : F(X) × F(X) → F(X) is defined, where F(X) denotes the set of all fuzzy subsets of X. Mapping ST,Inc maps two fuzzy sets A and B to a fuzzy set ST,Inc(A,B) in X called their fuzzy set of similarity. A measure of similarity between A and B is then obtained by applying the composite of fuzzy measure and fuzzy similarity mapping on the pair (A,B). Several properties of fuzzy set of similarity and the measure of fuzzy similarity are obtained within the framework of Lukasiewicz fuzzy implicator and its respective t-norm and t-conorm. Many examples of measure of similarity are also constructed.

Suitable interval-valued intuitionistic fuzzy topological spaces

2021 ◽  
pp. 1-11
Author(s):  
O.R. Sayed ◽  
N.H. Sayed ◽  
Gui-Xiu Chen
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Topological Spaces ◽  
Fuzzy Topology ◽  
Intuitionistic Fuzzy ◽  
Set Operations ◽  
The Family ◽  
Fuzzy Topological Spaces ◽  
Interval Valued

In the present paper, a characterization of the intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy sets and their set-operations are given. By making use of these characterizations, the relationships between the interval-valued intuitionistic fuzzy topology and four fuzzy topologies associated to it are studied. For this reason, some subclasses of the family of interval-valued intuitionistic fuzzy topologies on a set which we call pre-suitable and suitable are introduced. Furthermore, the concepts of homeomorphism functions and compactness in the framework of interval-valued intuitionistic fuzzy topological spaces are introduced and studied.

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