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This monograph is devoted to various types of algebras of functions with n variables. It is the first complete monograph (in English) on this topic, covering mainly the Russian literature. It is addressed to all algebraists working in the area of universal algebras, semigroup theory, etc. It is also a useful source of information for graduate and PhD students who are starting their research in this area.
The book is the first monograph in the 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 the Russian literature: In 2006, the first version of this book was edited in Russian and it is now presented in an extended version, where two new and very important chapters are added. The monograph is a broad survey of unknown or little-known Russian literature on algebras of multiplace functions and presents to the mathematical community a beautiful and strongly developing theory.
List of contents
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
About the author
Wieslaw A. Dudek, Politechnika Wroc¿awska, Wroc¿aw, Poland; Valentin S. Trokhimenko, Pedagogical University, Vinnitsa, Ukraine.
