# Advances in Fuzzy Implication Functions

• Michał Baczyński
• Gleb Beliakov
• Humberto Bustince Sola
Book

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 300)

1. Front Matter
Pages 1-6
2. Sebastià Massanet, Joan Torrens
Pages 1-30
3. Yun Shi, Bart Van Gasse, Etienne E. Kerre
Pages 31-51
4. Feng Qin, Michał Baczyński
Pages 53-72
Pages 73-99
6. Benjamín Bedregal, Gleb Beliakov, Humberto Bustince, Javier Fernández, Ana Pradera, Renata Reiser
Pages 101-124
7. Dana Hliněná, Martin Kalina, Pavol Král’
Pages 125-153
8. Józef Drewniak, Jolanta Sobera
Pages 155-176
9. Michał Baczyński, Balasubramaniam Jayaram
Pages 177-204
10. Back Matter
Pages 0--1

### Introduction

Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. These functions are not only fundamental for fuzzy logic systems, fuzzy control, approximate reasoning and expert systems, but they also play a significant role in mathematical fuzzy logic, in fuzzy mathematical morphology and image processing, in defining fuzzy subsethood measures and in solving fuzzy relational equations.

This volume collects 8 research papers on fuzzy implication functions.

Three articles focus on the construction methods, on different ways of generating new classes and on the common properties of implications and their dependencies. Two articles discuss implications defined on lattices, in particular implication functions in interval-valued fuzzy set theories. One paper summarizes the sufficient and necessary conditions of solutions for one distributivity equation of implication. The following paper analyzes compositions based on a binary operation * and discusses the dependencies between the algebraic properties of this operation and the induced sup-* composition. The last article discusses some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. These papers aim to present today’s state-of-the-art in this area.

### Keywords

Aggregation Operators Fuzzy Sets Implication Operators

### Editors and affiliations

• Michał Baczyński
• 1
• Gleb Beliakov
• 2
• Humberto Bustince Sola
• 3
• 4
1. 1., Institute of MathematicsUniversity of SilesiaKatowicePoland
2. 2.School of Information TechnologyDeakin UniversityBurwoodAustralia
3. 3.Departamento de Automática y, ComputaciónUniversidad Pública de NavarraPamplonaSpain
4. 4., Departamento de Ciencias de laUniversidad Rey Juan CarlosMostolesSpain

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-35677-3
• Copyright Information Springer-Verlag Berlin Heidelberg 2013
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages Engineering
• Print ISBN 978-3-642-35676-6
• Online ISBN 978-3-642-35677-3
• Series Print ISSN 1434-9922
• Series Online ISSN 1860-0808
• Buy this book on publisher's site