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Research and Exposition in Mathematics -- Volume 26

   Enlarged Picture

I. Chajda, G. Eigenthaler, H. Lšnger

Congruence Classes in Universal Algebra


x + 218 pages, soft cover, ISBN 3-88538-226-1, EUR 28.00, 2003

Congruence relations play an important role when investigating universal algebras. On the one hand, the structure of the congruence lattice of a given algebra reveals much information on the underlying algebra. On the other hand, via congruence relations quotient algebras can be formed which may have "nicer" properties than the original algebras. Moreover, in many cases congruences are determined by some of their classes. For instance in the case of groups, rings and Boolean algebras, congruences are determined by each single one of their classes. The aim of the present book is to present the most important results concerning congruence classes, dependences between them as well as connections to subalgebras. Thus the reader is informed on the developments in this field during the last decades.


List of Contents:

 
  Preface ix
 
  1: Preliminaries  
 
  2: Examples of algebraic structures 
2.1 Ordered structures 11
2.2 Groupoids and related algebras 19
2.3 Ring-like algebras 24
 
  3: Congruence classes and subalgebras 
3.1 Permutability 27
3.2 Distributivity and modularity 32
3.3 Direct decomposability 35
 
4: Congruence classes and subalgebras 
4.1 Maltsev description of congruence classes 39
4.2 Congruence classes which are subuniverses 40
4.3 Subuniverses which are congruence classes 41
4.4 Algebras having no proper subuniverse as a congruence class 43
4.5 Hamiltonian algebras 48
 
5: Extension properties of congruence and their classes 
5.1 Congruence extension property 51
5.2 Block extension property 54
 
  6: Regularity and its modifications 
6.1 Regularity 57
6.2 Transferable congruences 61
6.3 Regularity of single algebras 61
6.4 Weak regularity 64
6.5 Weakly regular algebras in varieties with principle compact congruences 68
6.6 Local regularity 71
6.7 Regularity with respect to unary terms 76
6.8 Subregularity 77
6.9 Dual regularity 78
6.10 Balanced algebras 80
 
  7: Coherence and its modifications 
7.1 Coherence 85
7.2 Weak coherence 87
7.3 Local coherence 90
7.4 t-Coherence 92
7.5 Uniformity 93
7.6 Consistent algebras 94
 
8: Local congruence conditions 
8.1 Permutability at 0 97
8.2 Distributivity at 0 99
8.3 Arithmeticity at 0 102
8.4 Modularity at 0 103
8.5 Weakly parallel classes 108
8.6 Decomposability of kernels 110
 
  9: Characterizations of congruence classes 
9.1 Congruence classes in regular and permutable varieties 111
9.2 Congruence classes in regular varieties 116
9.3 Congruence kernels 119
9.4 Deductive systems 120
9.5 Relative deductive systems 123
9.6 Convex sets 127
9.7 Congruence kernels in pseudocomplemented semilattices 134
 
  10: Ideals in universal algebras 
10.1 Ideals and ideal determined varieties 137
10.2 Ideals and congruence kernels 141
10.3 Finite bases for ideal terms 145
10.4 Ideal congruence properties 151
10.5 Ideals in locally regular and permutable at 0 varieties 155
10.6 Ideal extension property 159
10.7 Ideals, congruence kernels and tolerance kernels of lattices 160
 
  11: Directly decomposable congruence classes 
 
12: One-block congruences 
12.1 Semimodularity of one-block congruences 185
12.2 Rees algebras and Rees varieties 187
12.3 Rees ideal algebras and Rees ideal varieties 189
12.4 Rees sublattices of a lattice 192
 
Bibliography 197
 
Index 211