Algebras of Multiplace Functions

ISBN-13: 9783110269284 / Angielski / Twarda / 2012 / 399 str.

Wieslaw A. Dudek;Valentin S. Trokhimenko

Opis

This monograph is devoted to various types of algebras of functions with n variables. It is the first complete monograph (in English) on this area, covering mainly Russian literature.It aresses all algebraist working in the area of universal algebras, semigroup theory. Itis alsofruitful source for graduate and PhD students who are starting their research in this area. The book is the first monograph in English mathematical literature which provides readers with a very systematical study of the notion of Menger algebras and its generalizations and applications. The results presented here were originally published mostly in Russian literature: In 2006 the first version of this book was edited in Russian; now is presented an extended version. Two big new and very important chapters are added.The monograph is a very good survey of unknown or little-known Russian literature on algebras of multiplace functions. It uncovers to the mathematical community a beautiful and strongly developing theory.

This monograph is the first one in Englishmathematical literature which is devoted to the theory of algebras of functions of several variables. The book contains a comprehensive survey of main topics of this interesting theory. In particular the authors study the notion of Menger algebras and its generalizations in very systematic way. Readers are provided with complete bibliography as well as with systematic proofs of these results.

Spis treści

Designations

Introduction

1 Main concepts
1.1 Elements of the theory of relations
1.2 Functions and operations
1.3 Algebraic systems
1.4 Closure operations
1.5 Notes on Chapter 1

2 Menger algebras of functions
2.1 Definitions and fundamental notions
2.2 Menger semigroups
2.3 v-regular Menger algebras
2.4 i-solvable Menger algebras
2.5 Group-like Menger algebras
2.6 Antisymmetric Menger algebras
2.7 Representations of Menger algebras
2.8 Notes on Chapter 2

3 Ordered Menger algebras
3.1 Menger algebras of relations
3.2 F.o. and p.q-o. Menger algebras
3.3 Algebras of reversive functions
3.4 (f)-, (g)-, (f,g)-Menger algebras
3.5 Subtraction Menger algebras
3.6 Restrictive Menger algebras
3.7 Functional Menger systems
3.8 Notes on Chapter 3

4 Relations between functions
4.1 Stabilizers of Menger algebras
4.2 Stabilizers of functional Menger systems
4.3 Stationary subsets
4.4 Semi-compatibility relation
4.5 Co-definability relation
4.6 Connectivity relation
4.7 Projection equivalence relation
4.8 Semiadjacency relation
4.9 Notes on Chapter 4

5 (2, n)-semigroups of functions
5.1 (2, n)-semigroups and their representations
5.2 Menger (2, n)-semigroups
5.3 Projection relations on (2, n)-semigroups
5.4 Notes on Chapter 5

6 Systems of multiplace functions
6.1 Menger systems
6.2 Menger T-systems
6.3 Positional algebras
6.4 Mal'cev-Post iterative algebras
6.5 Semigroups of functions
6.6 Central semigroups of operations
6.7 Algebras of vector-valued functions
6.8 Notes on Chapter 6

7 Open problems
7.1 Closure operations
7.2 Menger algebras of functions
7.3 Menger algebras of relations
7.4 (2, n)-semigroups
7.5 Positional algebras

Bibliography

Index

O autorze

Wieslaw A. Dudek, Politechnika Wroc awska, Wroc aw, Poland; Valentin S. Trokhimenko, Pedagogical University, Vinnitsa, Ukraine.