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O'Donnell C.J. — Incidence Algebras
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Название: Incidence AlgebrasАвтор: O'Donnell C.J. Аннотация: While the incidence algebra of a locally finite partially ordered set X over a commutative ring /?, denoted 1{Х, Л), was introduced in the mid 1960's as a natural setting for the study of some combinatorial problems, it soon became apparent, that this algebra was an interesting object to study. In particular, it includes such standard ring theory examples as the product of copies of a ring R and the ring of upper triangular matrices over Л. The papers by Farkas Doubilet, Rota, and Stanlev and Leroux and Sarraille each investigated some algebraic aspects of incidence algebras and each also expressed the hope that their work would stimulate the interest of other researchers. In this direction, many properties of incidence algebras have been obtained, yet no single source indicative of the current level of knowledge on this subject is available. It is the purpose of the second part of this work (Chapters 4 through 8) to present some of the directions of development that have been explored. The book does not
Язык: Рубрика: Математика /Серия: Сделано в холле Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1997Количество страниц: 355Добавлена в каталог: 19.05.2011Операции: Положить на полку |
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141 226 227 189—194 197 198 200—202 217—224 227 228 230 189 190 218 2-subset 5—8 Algebra, anti-isomorphism 147 Algebra, boolean 3 42 47 49 71 91 110 144 165—167 175 189 190 196 199 202 219 225 226 Algebra, Boolean, complete 272 Algebra, dual 133 150 Algebra, embedding 171 186 187 Algebra, finite-dimensional 171 Algebra, isomorphism 15 16 19 24 27 28 32 53 56 77 111 113—117 119 120 133 136 137 157 170—173 186 187 237 261 263 279 299 Algebra, PI 297 298 Algebra, right regular representation 171 Algebra, subalgehra 19 28 31 Algebra, topological 10 18 Algebraic closure 44 87 Algebraic topology 50 Amitsur Levitski Theorem 299 Annihilator 95 242 Antichain 3 15 30 35 46 199 202 220 224—229 237—239 265 281 294 305 Antichain, maximal 220 225—227 259 294 Antipode 132 151 152 157—161 Antisymmetry 1 Associative law 17 121 atom 47 73—76 Augmentation 121 Bialgebra 130 131 133 136 140 141 143—146 149—152 157 159 Bialgebra, binomial 161 Bialgebra, Dirichlet 157 Bialgebra, Faa di Bruno 148 159 Bialgebra, hereditary 144 161 Block, partition 71—73 90 92 102 103 105 108 144 145 159 100 244 271 Cantor — Bendixson 228—230 Category 137—139 Category, isomorphism 139 140 Category, locally finite 137—139 Category, Moebius 18 Category, subcategory 138 139 Cayley — Hamilton theorem 23 24 Center, group 84 Center, incidence algebra 29—31 262 27 Chain 3 Chain, incomparable 219 221 245 Chain, length 3 5 11 37 38 158 161 174 233 240 245 246 293 300 Chain, maximal 46 73 74 113—115 158 Chain, of filters 166 Chain, of ideals 272 Chain, of ideals, prime 217 240 246—248 Characteristic zero 11 20 21 23 110 113 116 127 192 Chief series 82—84 Closed, element 45 46 Closed, map 198 208 219 Closed, set 195 196 198 202 208 209 Closure operator 44—46 52 Coalgebra 121—133 135—137 141 157 Coalgebra, binomial 132 133 Coalgebra, Boolean 132 141 Coalgebra, cleavage 141 Coalgebra, combinatorial 126 Coalgebra, Dirichlet 136 157 Coalgebra, divided powers 135 156 157 Coalgebra, divided powers, multivariate 135 Coalgebra, dual 135—137 157 Coalgebra, embedding 154 Coalgebra, Eulerian 137 144 Coalgebra, Faa di Bruno 145 146 Coalgebra, finite point lattice 141 Coalgebra, infinitesimal 141 Coalgebra, isomorphism 132 133 157 Coalgebra, Quotient 123 Coalgebra, subcoalgebra 126 127 Coassociative law 121 128 Coatom 47 65 76 Cobounded 174 175 187 189 190 192 193 199 220 222 223 227 Cocommutative 128 137 146 Coefficient, binomial 127 131 Coefficient, bisection 131 132 141 Coefficient, Faa di Bruno 145 Coefficient, Gaussian 68 136 137 Coefficient, incidence 109 111 140 Coefficient, q-binomial 68 116 Coefficient, section 127—129 131 132 136 141 145 Cofinite set 2 230 Coideal 122 124 129 130 138 140 COLOR 101 134 Coloring 101 105 134 Coloring, weight 101 102 104 105 Combinatorics 9 18 33 39 87 122 141 160 161 Compact 195 Compact, Hausdorff 228—230 Compact, Hausdorff, degree 228 229 Compact, Hausdorff, rank 228 229 Compact, regularly 293 296 Complement, of an element 165 167 Complement, set 230 245 Comultiplication 121 128 138 141 Conjugacy class 84 139 140 Connected component 9 15 29 263 264 Containment map 200 203 204 206—209 232 237 Containment set 203 205 214 Convex hull 4 23 24 193 220 Convex set 4 24 174 175 192 193 219 221 Convolution 150 151 153 156 157 159 Coset 103 215 Counitary product law 121 122 Counitmap 121 123 124 128 138 141 149 Countable intersection property 293 296 Cover 3 30 46 47 73—75 Cross cut 46—49 65 CYCLE 91—93 103—105 Cycle, disjoint 91—93 102 Cycle, length 91—93 DCC 306 de Bruijn's Theorem 102 105 Depth 294 Depth, finite 294 296 Derivation 141 251—256 259 Derivation, additive 253—258 Derivation, higher 259 Derivation, inner 252 255 256 259 Derived set 228 Diagonalization 121 123 129 130 149 Differentiation operator 161 Dirichlet, bialgebra 157 Dirichlet, coalgebra 136 157 Dirichlet, convolution 157 Dirichlet, series 28 157 Dirichlet, series, multivariate 136 Dual, algebra 133 150 Dual, coalgebra 135—137 157 Dual, lattice 47 49 Dual, poset 2 39 Dual, space, linear 125 e function 19 21 E-equivalence 19 24 Eificnspace 171 Eigenvalue 171 Enumeration problem 137 Equivalence class 1 4 9 19 20 24 26 101 102 104 105 107 129 129 285 301 Equivalence relation 1 4 9 19 25 26 107—110 112 115 129 133 136 138 148 153 211 285 301 Equivalence relation, compatible (on morphisms) 138 Equivalence relation, order compatible 19 20 23 25 107—109 112 128—130 132 133 136 141—144 148 149 151 152 155 Euler, characteristic 50—52 Euler, characteristic, reduced 50 Euler, phi function 103 Face 50 Field, dimension 50 Field, extension 44 87 Field, extension, degree 87 Field, extension, dimension 87 Field, finite 44 66—69 71 87 93 94 115 Field, order 44 69 87 93 94 Field, splitting 87 Field, subfield 44 87 224 Filter 165 Filter, bounded 199 220 227 Filter, prime 166 167 195 196 206 231—233 235 237 Filter, prime, minimal 166 234 235 Filter, principal 34 166 229 Filter, unbounded 199 220 223 227 Fixed point 202 Fully-nilpotent 210 236 Fully-nilpotent, on a set 173 Function, additive 252 Function, algebraic-tally bounded 189 202 217 Function, charactoisistic diagonal 29 Function, fractional 275 Function, Moebius 1 2 29 58 59 65 84 86 91 Function, multiplicative 278 279 Function, nilpotent 177 238 246 Function, potential 256 Function, unbounded 22 G-equivalence 10 Gib 163 164 Graph 9 18 134 Graph, planar 161 Graph, subgraph 135 Graph, theory 135 Graph, vertex 135 Group 18 82 84—273 275 Group, abelian 82—88 Group, abelian, elemental finite 70 Group, center 84 Group, centralizer 9 Group, cyclic 42 71 88 103 Group, direct product 70 83—85 Group, finite 82 83 139 Group, index 93 103 Group, isomorphism 84—86 156 157 159 273 279 Group, multiplicative 42 87 Group, nilpotent 83 Group, normalizer of a 84 Group, of multiplicative functions 156 157 159 Group, order 42 71 83 84 94 103 Group, permutation 101 Group, quotient 82 85 Group, simple 85 Group, simple, finite 84—86 Group, solvable 82 83 Group, solvable, finite 83 Group, subgroup 42 70 71 82—86 101 103 Group, subgroup, automorphism 273—275 Group, subgroup, commutator 80 82 Group, subgroup, complement 82 Group, subgroup, conjugate 102 Group, subgroup, maximal 82—84 Group, subgroup, normal 82—84 Group, subgroup, normal, minimal 82 83 Group, supplement 84 85 Group, Sylow 70 83 Group, symmetric 91 299 Hadamard product 25 26 32 107 252 274 Hasse diagram 3 5 9 22 164 257 Hasse diagram, vertex 3 9 Hausdorff 196 Height 115 293 294 Height, finite 294 295 Homeomorphism 197 198 208 218 221 224 228 229 231 234 238 247—249 Homology 18 Hopf algebra 132 140 151—153 156 161 Hopf algebra, Faa di Bruno 159 160 Hopf algebra, hereditary 153 Hopf algebra, isomorphism 157 Ideal map, maximal 211—215 Ideal map, prime 211 213 231 232 Ideal map, prime, minimal 238 Ideal, maximal 97 168 169 171 185 180 191 193 194 196—200 202 203 206 208 209 213 218 230 234 240—242 248 249 261 Ideal, maximally disjoint 182 234 241 246 247 Ideal, nil 180 181 184 185 306 Ideal, nilpotent 180—184 234—240 242 243 247—249 301 305 Ideal, of a poset 34 Ideal, of a poset, principal 34 Ideal, prime 179—183 185 189 194 195 200 202—204 206 209 210 217 218 220 222 223 230 237 239—244 246 Ideal, prime, minimal 217—219 223 227 228 230 231 234—236 239 249 Ideal, principal 36 306 Idempotent 86 260 261 263 266 272 280 306 Idempotent, central 263 Idempotent, orthogonal 172 261 266 267 278 280 Idempotent, orthogonally split 270 272 Idempotent, pairwise orthogonal 263 Idempotent, pairwise orthogonal primitive 263 Idempotent, primitive 263 Incidence algebra 10 Incidence algebra, algebraic 304 Incidence algebra, automorphism 35 274 278 Incidence algebra, automorphism, inner 35 273 278 279 Incidence algebra, automorphism, outer 281 Incidence algebra, automorphism, standard 279—281 Incidence algebra, basis 26 Incidence algebra, commutative 110 Incidence algebra, direct product 29 40 Incidence algebra, finite dimensional 173 Incidence algebra, free 148 Incidence algebra, invcrtible element of 23 27 33 Incidence algebra, isomorphism 40 260 261—263 266 268 270—273 281 Incidence algebra, PI 299 300 Incidence algebra, quotient 212 Incidence algebra, reduced 27 32 33 107 109—115 130 Incidence algebra, reduced, full binomial type 112—114 116 121 Incidence algebra, reduced, standard 25 26 28 110 116 117 119 120 Incidence algebra, snbalgebra 18—26 28 29 31 32 107 114 117 121 130 185 Incidence algebra, unit 289 291 292 Incidence bialgebra 141 142 153 Incidence bialgebra, embedding 153 Incidence bialgebra, free 149 157 Incidence bialgebra, reduced 132 141 143 144 151 Incidence bialgebra, reduced, free 156 Incidence bialgebra, reduced, standard 144 Incidence coalgebra 124—128 131 135 140—142 151 161 Incidence coalgebra, category 138 139 Incidence coalgebra, free 148 Incidence coalgebra, quotient 138 Incidence coalgebra, reduced 130 132 133 136 137 138 141 149 151 152 156 157 159 Incidence coalgebra, reduced, Boolean 132 Incidence coalgebra, reduced, free 148 Incidence coalgebra, reduced, inner 138—140 Incidence coalgebra, reduced, standard 130 133 135 136 138 139 144 145 149 151 153 157 159 Incidence coalgebra, reduced, strongly 138 139 Incidence coalgebra, subcoalgebra 122 126 127 Incidence Hopf algebra 153 154 157 Incidence Hopf algebra, free 157 Incidence Hopf algebra, reduced 143 152 156 Incidence Hopf algebra, reduced, free 156 Incidence ring isomorphism 262—264 271 Inclusion-Exclusion Principle 43 Indeterminate 27 113 128 146 147 297 Index of nilpotency 176 Integers, under divisibility 2 27 33 43 71 120 136 157 164 Integers, under usual ordering 1 5 18 26 37 41 42 71 127 135 157 164 288 Integral domain 120 174 187 240 243 249 Interval poset, the 48 52 Isomorphism problem 260 264 266 272 273 Join 73 75 108 163 164 Join of atoms 47 73—75 Join of atoms, irredundant decomposition 73—75 Join, semilattice 54—56 Jordan chain condition 73 74 113 Kernel 77 90 95 96 123 124 130 Kronecker delta 127 128
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