1 Introduction ................................................. 9
2 Algebras and classes of algebras ............................ 15
2.1 Identities and varieties of algebras ................... 16
2.2 Quasi-identities and quasi varieties of algebras ....... 18
2.3 Closure and kernel operators ........................... 19
2.4 The Galois Connection (Id,Mod) ......................... 20
2.5 Conjugate pairs of additive closure operators .......... 21
3 Hypersubstitutions .......................................... 23
3.1 Basic definitions ...................................... 23
3.2 Derived algebras of a given type ....................... 24
3.3 Solid varieties ........................................ 26
3.4 The closure operator D ................................. 27
3.5 The lattice of solid varieties of a given type τ ....... 28
3.6 Examples of solid varieties ............................ 28
3.7 The closure operators χA and χE ........................ 30
3.8 Fluid varieties ........................................ 32
3.9 The dimension .......................................... 36
3.10 Dimensions of varieties of lattices .................... 36
3.11 Dimensions of subvarieties of regular bands ............ 39
4 Solid quasivarieties ........................................ 48
4.1 Mal'cev type theorems .................................. 49
5 Hyperquasi-identities ....................................... 54
5.1 The closure operators χA χQE ............................ 55
6 Examples of hyperquasi-identities ........................... 59
6.1 Abelian algebras ....................................... 59
6.2 Solvable algebras ...................................... 60
6.3 Groupoids .............................................. 61
6.4 Modes .................................................. 62
6.5 Quasigroups ............................................ 62
6.6 Groups ................................................. 63
6.7 Unary algebras ......................................... 64
6.8 Semidistributive lattices .............................. 66
6.9 Distributive lattices .................................. 67
7 Hyperequational logic ....................................... 68
7.1 Hyperidentities and hypervarieties ..................... 68
7.2 Free solid algebras .................................... 68
7.3 Totally invariant congruences .......................... 69
7.4 Birkhoff's type theorems for hypervarieties ............ 70
8 Hyperquasi-equational logic ................................. 77
9 M-dimension of a variety .................................... 86
9.1 M-derived algebras ..................................... 86
9.2 M-solid algebras ....................................... 87
9.3 M-solid varieties ...................................... 88
9.4 M-dimension ............................................ 90
9.5 M-dimensions of varieties of lattices .................. 91
9.6 M-dimensions of varieties of regular bands ............. 93
10 Quasicompact classes ....................................... 101
11 M-solid quasivarieties ..................................... 102
12 M-hyperquasi-identities .................................... 106
12.1 Closure operators χAM χQEM.............................. 109
13 Examples of M-hyperquasi-identities ........................ 1ll
13.1 Quasigroups ........................................... 1ll
13.2 Distributive lattices ................................. 1ll
13.3 Boolean Algebras ...................................... 112
13.4 Unary algebras ........................................ 113
13.5 Flat algebras ......................................... 118
14 Solution of the hyperbasis problem ......................... 121
14.1 Solution of the hyperquasi basis problem .............. 122
14.2 Hyperbasis problem in varieties ....................... 123
15 Solution of the M-hyperbasis problem ....................... 125
15.1 Solution of the M-hyperquasi basis problem ............ 126
15.2 M-hyperbasis problem in varieties ..................... 127
16 M-quasicompact classes ..................................... 128
17 M-hyperquasi-equational logic .............................. 129
18 M-hyperequational logic .................................... 135
18.1 M-hyperidentities and M-hypervarieties ................ 135
18.2 M-derived classes ..................................... 135
18.3 Free M-solid algebras ................................. 136
18.4 M-totally invariant congruences ....................... 138
18.5 Birkhoff's theorems for M-hypervarieties .............. 138
19 Solution of the hyperequational problems ................... 144
20 Solution of liyperquasi-equational problems ................ 146
21 Summary in German - Zusammenfassung ........................ 148
22 Summary in Polish - Streszczenie ........................... 150
BIBLIOGRAPHY .................................................. 151
INDEX ......................................................... 164
|
