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Pitkethly J., Davey B.A. Ч Dualisability: Unary Algebras and Beyond
Pitkethly J., Davey B.A. Ч Dualisability: Unary Algebras and Beyond



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Ќазвание: Dualisability: Unary Algebras and Beyond

јвторы: Pitkethly J., Davey B.A.

јннотаци€:

Natural duality theory is one of the major growth areas within general algebra. This text provides a short path to the forefront of research in duality theory. It presents a coherent approach to new results in the area, as well as exposing open problems.
Unary algebras play a special role throughout the text. Individual unary algebras are relatively simple and easy to work with. But as a class they have a rich and complex entanglement with dualisability. This combination of local simplicity and global complexity ensures that, for the study of natural duality theory, unary algebras are an excellent source of examples and counterexamples.
A number of results appear here for the first time. In particular, the text ends with an appendix that provides a new and definitive approach to the concept of the rank of a finite algebra and its relationship with strong dualisability.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2005

 оличество страниц: 263

ƒобавлена в каталог: 18.11.2009

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
Abelian group      (see Group)
Algebraic constructions and dualisability, look up, Table      5 2 137
Algebraic constructions and non-dualisability, look up, Table      5 3 162
Algebraic, closure      219
Algebraic, operation      14
Algebraic, over      14
Algebraic, partial operation      14
Algebraic, relation      14
Algebraic, total operation      214
Almost equal      75
Alter ego      75 14Ч15
Alter ego of finite type      19
Alter ego, brute-force      21
Alter ego, purely relational      247
Alter ego, strong brute-force      224
Alternating chains and dualisability      53 172 183
Amalgamated union      139
Associate of a fragment      235 235Ч236
Associate of a fragment, compatible      236
Balanced      (see Hom-minimal)
Binary homomorphism      26 (see also Operation Boolean algebra)
Binary homomorphism, Stone duality      18
Boolean      136 151
Boolean p-semilattice      136 151
Boolean topology      16
Brute-force, alter ego      21
Brute-force, morphism      21
Brute-force, strong      224
Centre of a unary algebra      55
Chain of kernels      208
Chains of      53 172
Chains of enriched partial      213
Clones and dualisability      163Ч165
Closed substructures, generation of      223 226Ч227
Closure, algebraic      219
Closure, term      213 213Ч216
Closure, topological      219Ч220
column      35
Compactness, duality      20
Compatible associate for fragment      236
Congruence distributive, viii      246 245Ч247
Conjugation      70
Connected, component of a unary algebra      4 143
Connected, unary algebra      4
Connecting maps      218
Contagiously non-dualisable      183 183Ч185
Coordinate embedding      236
Coproduct of unary algebras      138Ч141 (see also Algebraic constructions disjoint
Coretraction      40
Cyclic element of a unar      86
Cyclic group      (see Group)
Decidability Problem for Dualisability      viii 209
Decidability Problem for Strong Dualisability      viii
Depth of a unary algebra      193
Directed ordered set      218
Disjoint union      138 141Ч149 162 187
Distance function on unary algebras      95
Distant union      765 165Ч169
Distinguish within      32
Distributive lattice      (see Lattice)
Dual of a structure      76
Dual of an algebra      76
Dual, category      15Ч16
Dualisability      viii 77 14Ч22
Dualisability and algebraic constructions, look up, Table      5 1 137
Dualisability and alternating chains      53 172 183
Dualisability and clones      163Ч165
Dualisability and finiteness      52
Dualisability, finite      27
Dualisability, full      22 22Ч23 89Ч91
Dualisability, inherent      779 201
Dualisability, Problem      iii 51
Dualisability, strong      23 23Ч24 89Ч91 211-212
Dualisability, within varieties      135Ч137 155
Dualisable      77
Dualisable, finitely      27
Dualisable, fully      22
Dualisable, inherently      779
Dualisable, strongly      23 (see also Dualisability)
Dualises      77
Dualises, strongly      214
Dualising structure      77
Duality      77 14Ч18
Duality, compactness      20
Duality, full      22 22Ч23 89Ч91
Duality, Hofmann Ч Mislove Ч Stralka      18
Duality, independence of generator for      19 23
Duality, infinite generators for      18Ч19
Duality, natural      vii 77
Duality, Pontryagin      18 79
Duality, Priestley      18
Duality, Stone      18
Duality, strong      23 23Ч24 89Ч91 211-212
Empty structure      15 20 214
Endomorphisms, cyclic group      40 47Ч48
Endomorphisms, group      47
Endomorphisms, lattice      27Ч28
Endomorphisms, semilattice      83
Enough algebraic operations      24 91
Enough algebraic operations and congruence distributivity      245Ч247
Enough algebraic operations and height      228Ч231
Enough algebraic operations and injectivity      91 245
Enough algebraic operations and rank      242Ч243
Enough algebraic operations and strong dualisability      24 231
Enough algebraic operations, proofs of      93 102 106 205
Enough algebraic operations, relative to an alter ego      229
Enough projections      243 243Ч247
Enriched partial clone      213
Entropic algebra      164 164Ч165 177
Equaliser      22
Evaluation      20
Evaluation, given by      27
Evaluation, locally an      27
Evaluation, natural embeddings given by      16 22
Fence in a unary algebra      95
Finite gentle basis      (see Gentle basis)
Finite support      (see support)
Finite Type Problem      27 149 209
Finite type, alter ego of      79
Finitely dualisable      27
Finitely separable      766 166Ч171
Finiteness and dualisability      52
Flat-semilattice operation      34 35 83
Fragment of a homomorphism      231
Fragment of a homomorphism, associate of      235 235Ч236
Fragment of a homomorphism, height of      232
Fred      (see Q)
Full dualisability      22 22Ч23 89Ч91
Full duality      22 22Ч23 89Ч91
Full versus Strong Problem      23 89Ч91 209
Fully dualisable      22 22Ч23 89Ч91
Generated by a finite algebra      2
Gentle, basis      62
Gentle, finite      64Ч68
Gentle, surjection      62
Ghost element of an algebra      22
Ghost element, applications of      80Ч86 151Ч154 160 177 181 184Ч185
Ghost element, method      21Ч22 80Ч81 151 153-154
Given by evaluation      21
Graphs and enough algebraic operations      228Ч231
Graphs and rank      240
Graphs and strong dualisability      226
Graphs at most      224 232
Graphs has a      224
Graphs of a fragment      232
Graphs of a homomorphism      224
Graphs of an algebra      224 225
Graphs, associated with a unary algebra      4 10Ч13 55 86 95 123 structure
Graphs, cyclic      18
Graphs, endomorphisms      47
Graphs, endomorphisms of      40 47Ч48
Graphs, Hanf number for dualisability      209
Graphs, height      223Ч241
Graphs, operation      46
Graphs, Pontryagin duality      18 19
Graphs, unbounded      224
Hofmann-Mislove-Stralka duality for semilattices      18
hold      200
Hom-minimal, algebra      74
Hom-minimal, locally      126
Hom-minimal, relation      120
Implication algebra, two-element      18 163Ч164 183Ч187
Independence of generator for duality      19 23
Infinite generators, duality for      18Ч19
Inherent Non-dualisability Problem      28
Inherently, dualisable      179 201
Inherently, non-dualisable      28 151Ч154 179 183
Injective, algebra      244
Injective, structure      23
Injectivity and strong dualisability      23 91 244Ч245 249
Inner element      61
Inverse limit      218
Inverse limit, applications of      218 221 238
Inverse limit, lemma      218Ч219
Inverse system      218
Irreducibility index      245
Isolated element      174
Isoreduct      90
Isoreduct, endomorphisms      27Ч28
Isoreduct, kernel of a unary algebra      52 (see also Zero-kernel one-kernel
Isoreduct, kernel unary algebra      52
Isoreduct, operations      27
Isoreduct, Priestley duality      18
Linear unary algebra      180 187Ч206 208
Locally an evaluation      21
Locally hom-minimal      126
Majority operation      83
Minimal algebra      191 197
Mono-unary algebra      (see Unarmorphism)
Mono-unary algebra in dual category      16
Mono-unary algebra, brute-force      21
Much ado about, nothing      20
Much ado about, nullaries      221
Natural duality      vii 17
Natural evaluation embeddings      16 22
Near-projection operation      39
Near-unanimity term, viii      25 247
Node of an ordered set      192
Non-dualisability      18 21Ч22 171
Non-dualisability and algebraic constructions, look up, Table      5 3 162
Non-dualisability, contagious      183 183Ч185
Non-dualisability, inherent      28 151Ч154 179 183
Non-dualisable      (see Non-dualisability)
Nullary, algebra      (see Zero-kernel unary algebra)
One-element algebra      2 3 20 214
One-kernel unary algebra      52 60Ч69 92Ч97 191Ч192 208
One-point extension      135 141 142Ч143 147 149
One-to-one away from      35
Operation, algebraic      14
Operation, flat      34 35
Operation, group      46
Operation, lattice      27
Operation, majority      83
Operation, near-projection      39
Operation, semilattice      83 154
Operation, total versus partial      15
Operations and dualisability      164
Operations of alter ego      214 221
Ordered set, associated with a unary algebra      188
Ordered set, unary algebra from      111Ч114
Outer element      61
P-semilattice      136 150Ч153
Partition, determined by an element      72 112
Partition, subuniverse determined by      123
Permutations, see zero-kernel unary, algebra petal      55 54Ч59
Pointed      176 (see also Disjoint union)
Pointed one-point extension      176
Pointwise extension of operations and relations      15
Pontryagin duality for abelian groups      19
Pontryagin duality for locally compact abelian groups      19
Pontryagin duality of given finite exponent      18
Priestley duality for distributive lattices      18
Pseudo-complemented      (see p-semilattice)
Pseudo-complemented semilattice      (see p-semilattice)
Purely relational alter ego      247
q      3 4 8 8Ч14 53 57 83 154Ч157 165 166 170 173 176Ч177
Quasi-equation      2
Quasi-injective      90Ч91 96 100 105 118
Quasi-ordered set associated with a unary algebra      95 188
Quasi-variety      1Ч3
Rank      212 236 236Ч249
Rank and congruence distributivity      245Ч247
Rank and enough algebraic operations      242Ч243
Rank and height      240
Rank and injectivity      244Ч245 249
Rank and strong dualisability      241
Rank at most      237
Rank of a partial operation      236 237 241
Rank of an algebra      237
Rank, infinity      237 249
Rank, one      241 243Ч249
Rank, two      241Ч249
Rank, zero      247Ч249
retract      40
Retraction      40
Retraction, finitely q-based      91
Retraction, fully dualisable      90
Retraction, gentle      62
Retraction, row      35
Retraction, sub      201
Retraction, term      40
Retraction, three-element unary algebras, dualisable      52Ч53
Retraction, unbiased      797 190Ч192
Satisfaction of      2
Satisfaction of a quasi-equation      2
Semilattice continued with unary operations      154
Semilattice continued, Hofmann Ч Mislove Ч Stralka duality      18
Semilattice, endomorphisms      83
Semilattice, flat      34 35
Semilattice, operation      83 154
Separable      166
Separable, finitely      166 166Ч171
Separated by homomorphisms      3
Separates      2
Small type      779 201
Stone duality for Boolean algebras      18
Strong brute-force alter ego      224
Strong dualisability      23 23Ч24 89Ч91 211Ч212
Strong dualisability and enough algebraic operations      24 231
Strong dualisability and height      226
Strong dualisability and injectivity      23 91 244Ч245 249
Strong dualisability and rank      212 241
Strong Dualisability Problem      viii
Strong duality      23 (see also Strong)
Strong idempotent      30 29Ч38
Strongly dualisable      23 (see also Strong dualisability)
Strongly dualises      214
Subreduct      183
Subretraction      201
Subuniverse determined by a partition      123
Support      27 57 143
Tame congruence theory      164
Term      40 (see also Algebraic constructions)
Term, closed      22 272 212Ч217
Term, closure      213 213Ч216
Term, isomorphic      766
Term, near-unanimity      viii 25 247
Term, retract      40 (see also Algebraic constructions)
Three-element      69Ч84 91 97Ч133
Three-kernel      52 86
1 2
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