Natural Dualities for the Working AlgebraistCambridge University Press, 12 nov. 1998 - 356 pages The theory of natural dualities, as presented in this text, is broad enough to encompass many known dualities through a rich assortment of substantive theorems yet concrete enough to be used to generate an array of previously undiscovered dualities. This text will serve as a user manual for algebraists, for category theorists and for those who use algebra in their work, particularly mathematicians and computer scientists interested in non-classical logics. It will also give the specialist a complete account of the foundations, leading to the research frontier of this rapidly developing field. As the first text devoted to the theory of Natural Dualities, it provides an efficient path through a large body of results, examples and applications in this subject which is otherwise available only in scattered research papers. To enable the book to be used in courses, each chapter ends with an extensive exercise set. Several fundamental unsolved problems are included. |
Table des matières
Dual Equivalences and Where to Find Them | 1 |
1 Categories | 2 |
2 Distributive lattices | 7 |
3 Quasivarieties | 15 |
4 Structured topological spaces | 20 |
5 Predualities | 29 |
Natural Dualities | 39 |
2 Duality theorems | 41 |
3 Piggyback dualities for varieties of Hey ting algebras | 199 |
4 Restricted Priestley dualities | 202 |
5 Piggyback dualities for varieties of Ockham algebras | 207 |
6 Piggyback dualities for varieties of distributive palgebras | 216 |
7 Completely dualisable quasivarieties | 221 |
8 Endodualisable algebras | 225 |
Optimal Dualities and Entailment | 233 |
1 Test algebrasschizophrenia strikes again | 234 |
3 Taming brute force with nearunanimity | 52 |
4 Refining a duality via entailment | 55 |
3 Strong Dualities | 62 |
1 Full duality and the dual category | 63 |
2 Strong duality and the role of injectivity | 71 |
3 Producing strong dualities | 79 |
Examples of Strong Dualities | 94 |
2 Arithmetical strong dualities | 98 |
3 NU strong dualities | 102 |
4 Twoforone strong dualities | 113 |
5 Vagrant dualities | 116 |
Sample Applications | 124 |
2 Endomorphism monoids and upside down lattices | 125 |
3 Theory of algebraic and existential closure | 129 |
4 Examples of algebraic and existential closure | 135 |
5 Injectives weak injectives and absolute subretracts | 144 |
6 Category equivalences | 159 |
What Makes a Duality Useful? | 171 |
1 Total structures and injectivity | 173 |
2 Unary structures coproducts and nearunanimity | 177 |
3 Better coproducts and logarithmicity | 182 |
4 Exclusively unary structures and arithmeticity | 186 |
Piggyback Dualities | 192 |
2 Piggyback dualities for distributivelatticebased algebras | 195 |
2 Entailment closure and the topology of failsets | 238 |
3 Failsets and optimal dualities | 240 |
4 Kleene algebras once again | 246 |
5 The structure of failsets | 251 |
6 Optimal dualities for varieties of distributive palgebras | 261 |
Completeness Theorems for Entailment | 270 |
2 Completeness of constructs for entailment | 274 |
3 Local entailment | 278 |
4 Optimal strong dualities and homentailment | 281 |
the threeelement chain | 286 |
Dualisable Algebras | 291 |
2 Nearunanimity as an obstacle to duality | 292 |
3 Some applications of the NU obstacle theorems | 296 |
4 The balanced subalgebra method | 299 |
5 The ghost element method | 304 |
6 The termclosed subset method | 309 |
7 Dualisable clones on a twoelement set | 317 |
Algebras | 325 |
Boolean Spaces | 337 |
341 | |
Notation | 348 |
350 | |
Expressions et termes fréquents
algebraic relation binary relations Boolean algebra Boolean space bounded Priestley space Chapter clone closed substructure closure congruence distributive Consequently constructs contains coproduct Davey define denote distributive lattice dom(h dual adjunction dual category dual equivalence dualisable element embedding endodualisable example Exercise existentially closed finite algebra finite subset following are equivalent full duality functor globally minimal failset graph GUHUR entails Hence Heyting algebra hom-entailment homomorphism implies injective intersection IS,P+ isomorphism ISPM Kleene algebras Lemma Lemma Let monoid morphism n-ary natural duality near-unanimity term non-dualisable non-extendable partial endomorphisms obtained Ockham space optimal duality partial operations Piggyback Duality preserves Priestley space projection prove quasi-variety retract S-ary term functions satisfies semilattice Stone algebras Strong Duality Theorem Structure Theorem subalgebra subdirectly irreducible surjective term-closed topological space topology total structure two-element unary variety whence yields a duality yields a strong