
Preface 
ix 




1: Preliminaries 





2: Examples of algebraic structures  
2.1 
Ordered structures 
11 
2.2 
Groupoids and related algebras 
19 
2.3 
Ringlike 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 
tCoherence 
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: Oneblock congruences  
12.1 
Semimodularity of oneblock 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 