Fuzzy Semigroups
Published on 10. July 2003
Book
Hardback
IX, 319 pages
978-3-540-03243-4 (ISBN)
Description
Lotfi Zadeh introduced the notion of a fuzzy subset of a set in 1965. Ris seminal paper has opened up new insights and applications in a wide range of scientific fields. Azriel Rosenfeld used the notion of a fuzzy subset to put forth cornerstone papers in several areas of mathematics, among other discplines. Rosenfeld is the father of fuzzy abstract algebra. Kuroki is re sponsible for much of fuzzy ideal theory of semigroups. Others who worked on fuzzy semigroup theory, such as Xie, are mentioned in the bibliogra phy. The purpose of this book is to present an up to date account of fuzzy subsemigroups and fuzzy ideals of a semigroup. We concentrate mainly on theoretical aspects, but we do include applications. The applications are in the areas of fuzzy coding theory, fuzzy finite state machines, and fuzzy languages. An extensive account of fuzzy automata and fuzzy languages is given in [100]. Consequently, we only consider results in these areas that have not appeared in [100] and that pertain to semigroups. In Chapter 1, we review some basic results on fuzzy subsets, semigroups, codes, finite state machines, and languages. The purpose of this chapter is to present basic results that are needed in the remainder of the book. In Chapter 2, we introduce certain fuzzy ideals of a semigroup, namely, fuzzy two-sided ideals, fuzzy bi-ideals, fuzzy interior ideals, fuzzy quasi ideals, and fuzzy generalized bi-ideals.
More details
Series
Edition
2003 ed.
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Illustrations
1 s/w Abbildung
IX, 319 p. 1 illus.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 23 mm
Weight
664 gr
ISBN-13
978-3-540-03243-4 (9783540032434)
DOI
10.1007/978-3-540-37125-0
Schweitzer Classification
Other editions
Additional editions
John N. Mordeson | Davender S. Malik | Nobuaki Kuroki
Fuzzy Semigroups
Book
12/2010
Springer
€171.19
Shipment within 7-9 days
Persons
Dr. Sunil Mathew is currently a Faculty Member in the Department of Mathematics, NIT Calicut, India. He has acquired his masters from St. Joseph's College Devagiri, Calicut, and Ph.D. from National Institute of Technology Calicut in the area of Fuzzy Graph Theory. He has published more than 75 research papers and written two books. He is a member of several academic bodies and associations. He is editor and reviewer of several international journals. He has an experience of 20 years in teaching and research. His current research topics include fuzzy graph theory, bio-computational modeling, graph theory, fractal geometry, and chaos.
Dr. John N. Mordeson is Professor Emeritus of Mathematics at Creighton University. He received his B.S., M.S., and Ph.D. from Iowa State University. He is a Member of Phi Kappa Phi. He is the President of the Society for Mathematics of Uncertainty. He has published 15 books and 200 journal articles. He is on the editorial board of numerous journals. He has served as an external examiner of Ph.D. candidates from India, South Africa, Bulgaria, and Pakistan. He has refereed for numerous journals and granting agencies. He is particularly interested in applying mathematics of uncertainty to combat the problem of human trafficking.
Dr. Davender S. Malik is a Professor of Mathematics at Creighton University. He received his Ph.D. from Ohio University and has published more than 55 papers and 18 books on abstract algebra, applied mathematics, graph theory, fuzzy automata theory and languages, fuzzy logic and its applications, programming, data structures, and discrete mathematics.
Content
1 Introduction.- 1.1 Notation.- 1.2 Relations.- 1.3 Functions.- 1.4 Fuzzy Subsets.- 1.5 Semigroups.- 1.6 Codes.- 1.7 Finite-State Machines.- 1.8 Finite-State Automata.- 1.9 Languages and Grammars.- 1.10 Nondeterministic Finite-State Automata.- 1.11 Relationships Between Languages and Automata.- 2 Fuzzy Ideals.- 2.1 Introduction.- 2.2 Ideals in Semigroups.- 2.3 Fuzzy Ideals in Semigroups.- 2.4 Fuzzy Bi-ideals in Semigroups.- 2.5 Fuzzy Interior Ideals in Semigroups.- 2.6 Fuzzy Quasi-ideals in Semigroups.- 2.7 Fuzzy Generalized Bi-ideals in Semigroups.- 2.8 Fuzzy Ideals Generated by Fuzzy Subsets of Semigroups.- 3 Regular Semigroups.- 3.1 Regular Semigroups.- 3.2 Completely Regular Semigroups.- 3.3 Intra-regular Semigroups.- 3.4 Semisimple Semigroups.- 3.5 On Fuzzy Regular Subsemigroups of a Semigroup.- 3.6 Fuzzy Weakly Regular Subsemigroups.- 3.7 Fuzzy Completely Regular and Weakly Completely Regular Subsemigroups.- 3.8 Weakly Regular Semigroups.- 4 Semilattices of Groups.- 4.1 A Semilattice of Left (Right) Simple Semigroups.- 4.2 A Semilattice of Left (Right) Groups.- 4.3 A Semilattices of Groups.- 4.4 Fuzzy Normal Semigroups.- 4.5 Convexity and Green's Relations.- 4.6 The Compact Convex Set of Fuzzy Ideals.- 4.7 Fuzzy Ideals and Green's Relations.- 5 Fuzzy Congruences on Semigroups.- 5.1 Fuzzy Congruences on a Semigroup.- 5.2 Fuzzy Congruences on a Group.- 5.3 Fuzzy Factor Semigroups.- 5.4 Homomorphism Theorems.- 5.5 Idempotent-separating Fuzzy Congruences.- 5.6 Group Fuzzy Congruences.- 5.7 The Lattice of Fuzzy Congruence Relations on a Semigroup.- 5.8 Fuzzy Congruence Pairs of Inverse Semigroups.- 5.9 Fuzzy Rees Congruences on Semigroups.- 5.10 Additional Fuzzy Congruences on Semigroups.- 6 Fuzzy Congruences on T*-pure Semigroups.- 6.1 T*-pure Semigroups.- 6.2Semilattice Fuzzy Congruences.- 6.3 Group Fuzzy Congruences.- 7 Prime Fuzzy Ideals.- 7.1 Preliminaries.- 7.2 Prime Fuzzy Ideals.- 7.3 Weakly Prime Fuzzy Ideals.- 7.4 Completely Prime and Weakly Completely Prime Fuzzy Ideals.- 7.5 Relationships.- 7.6 Types of Prime Fuzzy Left Ideals.- 7.7 Prime Fuzzy Left Ideals.- 7.8 Fuzzy m-systems and Quasi-prime Fuzzy Left Ideals.- 7.9 Weakly Quasi-prime Fuzzy Left Ideals.- 7.10 Fuzzy Ideals i(f) and I(f).- 7.11 Strongly Semisimple Semigroups.- 7.12 Fuzzy Multiplication Semigroups.- 7.13 Properties of Fuzzy Multiplication Semigroups.- 7.14 Fuzzy Ideal Extensions.- 7.15 Prime Fuzzy Ideals.- 8 Fuzzy Codes on Free Monoids.- 8.1 Fuzzy Codes.- 8.2 Prefix Codes.- 8.3 Maximal Fuzzy Prefix Codes.- 8.4 Algebraic Properties of Fuzzy Prefix Codes on a Free Monoid.- 8.5 Fuzzy Prefix Codes Related to Fuzzy Factor Theorems.- 8.6 Equivalent Depictions of Fuzzy Codes.- 8.7 Fuzzy Codes and Fuzzy Submonoids.- 8.8 An Algorithm of test for Fuzzy Codes.- 8.9 Measure of a Fuzzy Code.- 8.10 Code Theory and Fuzzy Subsemigroups.- 8.11 Construction of Examples by Closure Systems.- 8.12 Examples by *-morphisms.- 9 Generalized State Machines.- 9.1 T-generalized State Machines.- 9.2 T-generalized Transformation Semigroups.- 9.3 Coverings.- 9.4 Direct Products.- 9.5 Decompositions of T-generalized Transformation Semigroups.- 9.6 On Proper Fuzzification of Finite State Machines.- 9.7 Generalized Fuzzy Finite State Machines.- 9.8 Fuzzy Relations and Fuzzy Finite State Machines.- 9.9 Completion of Fuzzy Finite State Machines.- 9.10 Generalized State Machines and Homomorphisms.- 10 Regular Fuzzy Expressions.- 10.1 Regular Fuzzy Expressions.- 10.2 Codes Over Languages.- 10.3 Regulated Codes and Fuzzy Grammars.- References.