This monograph is the first one in English mathematical 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.
Author(s): Wieslaw A. Dudek, Valentin S. Trokhimenko
Publisher: de Gruyter
Year: 2012
Language: English
Pages: 401
Preface ......Page 6
1.1 Elements of theory of relations ......Page 12
1.2 Functions and operations ......Page 16
1.3 Algebraic systems ......Page 19
1.4 Closure operations ......Page 21
1.5 Notes on Chapter 1 ......Page 31
2.1 Definitions and fundamental notions ......Page 32
2.2 Menger semigroups ......Page 47
2.3 v-regular Menger algebras ......Page 53
2.4 i-solvable Menger algebras ......Page 61
2.5 Group-like Menger algebras ......Page 68
2.6 Antisymmetric Menger algebras ......Page 79
2.7 Representations of Menger algebras ......Page 87
2.8 Notes on Chapter 2 ......Page 93
3.1 Menger algebras of relations ......Page 96
3.2 F.o. and p.q-o. Menger algebras ......Page 101
3.3 Algebras of reversive functions ......Page 107
3.4 (⋏)-, (⋎)-, (⋏, ⋎)-Menger algebras......Page 114
3.5 Subtraction Menger algebras ......Page 130
3.6 Restrictive Menger algebras ......Page 146
3.7 Functional Menger systems ......Page 155
3.8 Notes on Chapter 3 ......Page 162
4.1 Stabilizers of Menger algebras ......Page 164
4.2 Stabilizers of functional Menger systems ......Page 178
4.3 Stationary subsets ......Page 189
4.4 Semi-compatibility relation ......Page 207
4.5 Co-definability relation ......Page 217
4.6 Connectivity relation ......Page 223
4.7 Projection equivalence relation ......Page 229
4.8 Semiadjacency relation ......Page 235
4.9 Notes on Chapter 4 ......Page 240
5.1 (2, n)-semigroups and their representations ......Page 241
5.2 Menger (2, n)-semigroups ......Page 259
5.3 Projection relations on (2, n)-semigroups ......Page 275
5.4 Notes on Chapter 5 ......Page 298
6.1 Menger systems ......Page 299
6.2 Menger T-systems ......Page 313
6.3 Positional algebras ......Page 321
6.4 Mal’cev-Post iterative algebras ......Page 332
6.5 Semigroups of functions ......Page 352
6.6 Central semigroups of operations ......Page 360
6.7 Algebras of vector-valued functions ......Page 363
6.8 Notes on Chapter 6 ......Page 372
7.2 Menger algebras of functions ......Page 374
7.4 (2, n)-semigroups ......Page 375
7.5 Systems of multiplace functions ......Page 376
Bibliography ......Page 377
Index of notations ......Page 393
Index ......Page 397
