1 Introduction ................................................. 6
2 Algebras ..................................................... 7
3 Equivalence relations and quotients .......................... 9
4 Varieties and identities .................................... 10
4.1 Free algebras .......................................... 10
4.2 Fully invariant congruences ............................ 13
4.3 Birkhoff's theorems for varieties ...................... 13
5 Graphical proofs ............................................ 19
5.1 Tree identities ........................................ 19
5.2 Tree identities in graphic ............................. 20
5.3 Birkhoff's derivation rules in graphic ................. 26
6 Motivation for derived algebras ............................. 33
7 Extensions of varieties ..................................... 34
8 A-identities ................................................ 36
8.1 Normal and regular identities .......................... 36
8.2 Sheffer and Peirce algebras ............................ 37
9 Inflations of semigroups .................................... 38
10 Normal identities ........................................... 39
11 The operator N .............................................. 41
11.1 Properties of normal identities ........................ 42
11.2 Not-normal algebras .................................... 43
11.3 G. Birkhoff's theorems for normal shifts ............... 46
11.4 The lattice L(Mod(N(V))) ............................... 48
12 Externary compatible identities ............................. 48
12.1 Externary compatible identities in lattices ............ 49
13 The operator E .............................................. 50
13.1 Externary compatible identities in groups .............. 54
14 Semilattice-ordered systems of algebras ..................... 56
14.1 Clifford constructions ................................. 57
14.2 Plonka sums ............................................ 59
14.3 Examples ............................................... 59
14.4 Definitions and properties ............................. 65
14.5 Regular consequences and proofs ........................ 67
15 The operator R .............................................. 72
15.1 Representation theorems for regular shifts ............ 72
15.2 Regular subdirectly irreducible algebras .............. 77
15.3 Jónsson's Lemma for regular shifts .................... 77
15.4 Free algebras in Mod(R(V)) ............................ 79
15.5 Regular congruences ................................... 81
15.6 Birkhoff's theorems for regular shifts ................ 82
15.7 The lattice L(Mod(R(V))) .............................. 83
15.8 EIS property, amalgamation in Mod(R(V)) ............... 85
15.9 Bases, finite bases, finite basis property ............ 86
15.10 The word problems for shifts .......................... 88
15.11 A-shifts of varieties ................................. 88
16 Selfdual varieties .......................................... 89
16.1 Selfdual algebras ...................................... 90
16.2 Selfdual congruences ................................... 92
16.3 Birkhoff's theorems for selfdual varieties ............. 93
17 M-hyperidentities and M-hypervarieties ...................... 96
17.1 M-hypersubstitutions ................................... 97
17.2 Monounary algebras ..................................... 98
17.3 M-solid algebras ....................................... 98
17.4 Unary algebras ......................................... 99
17.5 M-derived algebras ..................................... 99
17.6 Free M-solid algebras ................................. 100
17.7 M-totally invariant congruences ....................... 102
17.8 Birkhoff's theorems for M-hypervarieties .............. 102
18 Coalgebras ................................................. 108
18.1 Deterministic coalgebras .............................. 108
19 Coterms without variables .................................. 112
20 Coterms over one variable .................................. 117
21 Congruences in coalgebras .................................. 118
21.1 Quotient coalgebras ................................... 119
21.2 Nuclear congruences in coalgebras ..................... 119
22 Coidentities of one variable ............................... 121
23 Coterms over a set X of variables .......................... 121
24 Coidentities over a set X .................................. 123
25 Satisfaction of coidentities ............................... 123
26 Free deterministic coalgebras .............................. 123
27 M-cohypersubstitutions ..................................... 126
28 M-cohypervarieties ......................................... 127
29 Applications ............................................... 136
29.1 Boolean identities .................................... 136
29.2 Boolean hyperidentities ............................... 138
30 Pseudovarieties ............................................ 141
30.1 Operators on pseudodovarieties ........................ 141
30.2 The lattice of subpseudovarieties ..................... 146
31 Regular pseudovarieties .................................... 147
31.1 Jonsson's Lemma for regular shifts of
pseudovarieties ....................................... 148
31.2 The lattice of subpseudovarieties of R(V) ............. 149
32 M-solid pseudovarieties .................................... 153
33 Double systems of algebras ................................. 155
34 Generalizations ............................................ 159
34.1 Graphs ................................................ 159
34.2 Relational systems .................................... 159
34.3 Solid pseudovarieties ................................. 160
35 SUMMARY - STRESZCZENIE ..................................... 161
BIBLIOGRAPHY .................................................. 163
INDEX ......................................................... 185
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