Research and Exposition in Mathematics -- Volume 26

   Enlarged Picture

I. Chajda, G. Eigenthaler, H. Länger

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