M-Solid Varieties of Algebras Book

Preface vii
1 Basic Concepts 1
1.1 Subalgebras and Homomorphic Images 1
1.2 Direct and Subdirect Products 11
1.3 Term Algebras, Identities, Free Algebras 16
1.4 The Galois Connection (Id, Mod) 24
2 Closure Operators and Lattices 29
2.1 Closure Operators and Kernel Operators 29
2.2 Complete Sublattices of a Complete Lattice 31
2.3 Galois Connections and Complete Lattices 34
2.4 Galois Closed Subrelations 37
2.5 Conjugate Pairs of Additive Closure Operators 41
3 M-Hyperidentities and M-solid Varieties 49
3.1 M-Hyperidentities 49
3.2 The Closure Operators X[subscript M superscript A], X[subscript M superscript E] 62
3.3 M-Solid Varieties and their Characterization 64
3.4 Subvariety Lattices and Monoids of Hypersubstitutions 67
3.5 Derivation of M-Hyperidentities 71
4 Hyperidentities and Clone Identities 77
4.1 Menger Algebras of Rank n 78
4.2 The Clone of a Variety 84
5 Solid Varieties of Arbitrary Type 87
5.1 Rectangular Algebras 87
5.2 Solid Chains 96
6 Monoids of Hypersubstitutions 103
6.1 Basic Definitions 103
6.2 Injective and Bijective Hypersubstitutions 109
6.3 Finite Monoids of Hypersubstitutions of Type (2) 114
6.4 The Monoid of all Hypersubstitutions of Type (2) 119
6.5 Green's Relations on Hyp(2) 126
6.6 Idempotents in Hyp(2, 2) 138
6.7 The Order of Hypersubstitutions of Type (2, 2) 151
6.8 Green's Relations in Hyp(n, n) 171
6.9 The Monoid of Hypersubstitutions of Type (n) 181
6.10 Left-Seminearrings of Hypersubstitutions 184
7 M-Solid Varieties of Semigroups 197
7.1 Basic Concepts on M-Solid Varieties of Semigroups 197
7.2 Regular-solid Varieties of Semigroups206
7.3 Solid Varieties of Semigroups 252
7.4 Pre-solid Varieties of Semigroups 255
7.5 Locally Finite and Finitely Based M-solid Varieties 260
8 M-solid Varieties of Semirings 267
8.1 Necessary Conditions for Solid Varieties of Semirings 267
8.2 The Minimal Solid Variety of Semirings 269
8.3 The Greatest Solid Variety of Semirings 271
8.4 The Lattice of all Solid Varieties of Semirings 275
8.5 Generalization of Normalizations 283
8.6 All Pre-solid Varieties of Semirings 298 Bibliography 321 Glossary 331 Index 337