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| Pitkethly J., Davey B.A. — Dualisability: Unary Algebras and Beyond |
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| Предметный 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 14—15
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 235 235—236
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 163—165
Closed substructures, generation of 223 226—227
Closure, algebraic 219
Closure, term 213 213—216
Closure, topological 219—220
column 35
Compactness, duality 20
Compatible associate for fragment 236
Congruence distributive, viii 246 245—247
Conjugation 70
Connected, component of a unary algebra 4 143
Connected, unary algebra 4
Connecting maps 218
Contagiously non-dualisable 183 183—185
Coordinate embedding 236
Coproduct of unary algebras 138—141 (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 138 141—149 162 187
Distance function on unary algebras 95
Distant union 765 165—169
Distinguish within 32
Distributive lattice (see Lattice)
Dual of a structure 76
Dual of an algebra 76
Dual, category 15—16
Dualisability viii 77 14—22
Dualisability and algebraic constructions, look up, Table 5 1 137
Dualisability and alternating chains 53 172 183
Dualisability and clones 163—165
Dualisability and finiteness 52
Dualisability, finite 27
Dualisability, full 22 22—23 89—91
Dualisability, inherent 779 201
Dualisability, Problem iii 51
Dualisability, strong 23 23—24 89—91 211-212
Dualisability, within varieties 135—137 155
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 14—18
Duality, compactness 20
Duality, full 22 22—23 89—91
Duality, Hofmann — Mislove — Stralka 18
Duality, independence of generator for 19 23
Duality, infinite generators for 18—19
Duality, natural vii 77
Duality, Pontryagin 18 79
Duality, Priestley 18
Duality, Stone 18
Duality, strong 23 23—24 89—91 211-212
Empty structure 15 20 214
Endomorphisms, cyclic group 40 47—48
Endomorphisms, group 47
Endomorphisms, lattice 27—28
Endomorphisms, semilattice 83
Enough algebraic operations 24 91
Enough algebraic operations and congruence distributivity 245—247
Enough algebraic operations and height 228—231
Enough algebraic operations and injectivity 91 245
Enough algebraic operations and rank 242—243
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 243 243—247
Enriched partial clone 213
Entropic algebra 164 164—165 177
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 766 166—171
Finiteness and dualisability 52
Flat-semilattice operation 34 35 83
Fragment of a homomorphism 231
Fragment of a homomorphism, associate of 235 235—236
Fragment of a homomorphism, height of 232
Fred (see Q)
Full dualisability 22 22—23 89—91
Full duality 22 22—23 89—91
Full versus Strong Problem 23 89—91 209
Fully dualisable 22 22—23 89—91
Generated by a finite algebra 2
Gentle, basis 62
Gentle, finite 64—68
Gentle, surjection 62
Ghost element of an algebra 22
Ghost element, applications of 80—86 151—154 160 177 181 184—185
Ghost element, method 21—22 80—81 151 153-154
Given by evaluation 21
Graphs and enough algebraic operations 228—231
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 10—13 55 86 95 123 structure
Graphs, cyclic 18
Graphs, endomorphisms 47
| Graphs, endomorphisms of 40 47—48
Graphs, Hanf number for dualisability 209
Graphs, height 223—241
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 18 163—164 183—187
Independence of generator for duality 19 23
Infinite generators, duality for 18—19
Inherent Non-dualisability Problem 28
Inherently, dualisable 179 201
Inherently, non-dualisable 28 151—154 179 183
Injective, algebra 244
Injective, structure 23
Injectivity and strong dualisability 23 91 244—245 249
Inner element 61
Inverse limit 218
Inverse limit, applications of 218 221 238
Inverse limit, lemma 218—219
Inverse system 218
Irreducibility index 245
Isolated element 174
Isoreduct 90
Isoreduct, endomorphisms 27—28
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 180 187—206 208
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 21—22 171
Non-dualisability and algebraic constructions, look up, Table 5 3 162
Non-dualisability, contagious 183 183—185
Non-dualisability, inherent 28 151—154 179 183
Non-dualisable (see Non-dualisability)
Nullary, algebra (see Zero-kernel unary algebra)
One-element algebra 2 3 20 214
One-kernel unary algebra 52 60—69 92—97 191—192 208
One-point extension 135 141 142—143 147 149
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 111—114
Outer element 61
P-semilattice 136 150—153
Partition, determined by an element 72 112
Partition, subuniverse determined by 123
Permutations, see zero-kernel unary, algebra petal 55 54—59
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 8—14 53 57 83 154—157 165 166 170 173 176—177
Quasi-equation 2
Quasi-injective 90—91 96 100 105 118
Quasi-ordered set associated with a unary algebra 95 188
Quasi-variety 1—3
Rank 212 236 236—249
Rank and congruence distributivity 245—247
Rank and enough algebraic operations 242—243
Rank and height 240
Rank and injectivity 244—245 249
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 241 243—249
Rank, two 241—249
Rank, zero 247—249
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 52—53
Retraction, unbiased 797 190—192
Satisfaction of 2
Satisfaction of a quasi-equation 2
Semilattice continued with unary operations 154
Semilattice continued, Hofmann — Mislove — Stralka duality 18
Semilattice, endomorphisms 83
Semilattice, flat 34 35
Semilattice, operation 83 154
Separable 166
Separable, finitely 166 166—171
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 23—24 89—91 211—212
Strong dualisability and enough algebraic operations 24 231
Strong dualisability and height 226
Strong dualisability and injectivity 23 91 244—245 249
Strong dualisability and rank 212 241
Strong Dualisability Problem viii
Strong duality 23 (see also Strong)
Strong idempotent 30 29—38
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 22 272 212—217
Term, closure 213 213—216
Term, isomorphic 766
Term, near-unanimity viii 25 247
Term, retract 40 (see also Algebraic constructions)
Three-element 69—84 91 97—133
Three-kernel 52 86
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