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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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Krull dimension 240 243 244 246 247 Krull dimension, finite 240 241 243 247 Krull dimension, zero 240—244 247—249 Lagrange inversion formula 159—161 Lagrange inversion formula, multivariate 161 Large element 226 227 Lattice 164 Lattice, characteristic 81 Lattice, complete 165 Lattice, direct product 83 Lattice, distributive 3 61 76 77 79 81 126 127 164—166 195 196 204 Lattice, distributive, finite 61 62 Lattice, finite 46 47 49 57 61—66 69 73 Lattice, geometric 73 74 76 86 Lattice, geometric, finite 73 75 76 100 Lattice, geometric, rank 100 Lattice, isomorphism 44 72 204 238 Lattice, of partitions 71 72 86 90 102 Lattice, of subgroups 71 82 83 86 102 Lattice, of submodules 71 Lattice, of vector subspaces 71 Lattice, rank 75 76 Lattice, theory 163 168 Limit point 228 Linear algebra 136 171 Linear transformation 93—95 122—125 150 151 171 Linear transformation, diagonalizable 171 172 Linear transformation, invertible 95 Linear transformation, pairwise commuting 171 172 Linear transformation, scalar 171 Logarithm 88 Lower bound 163 164 Lub 163 164 Matrix 13 16 58—60 62—64 93 101 173 230 299 Matrix, column 60 Matrix, column, operation 60 Matrix, determinant 23 58—60 63 64 Matrix, diagonalizable 62 63 171 172 Matrix, idempotent 172 Matrix, invertible 24 60 63 173 Matrix, lower triangular 13 32 Matrix, nonsingular 93 172 Matrix, representation 171 Matrix, row 93 Matrix, subalgebra 13 32 171 172 299 Matrix, upper triangular 13 23 31 230 Matrix, upper triangular, strictly 230 Meet 65 163 164 Meet of coatoms 47 Meet, semilattice 54—56 Meet, semilattice, finite 57 59 Module, basis 53 60 302 Module, cyclic 69 94—96 98 Module, dimension 99 100 Module, direct sum of 94 100 Module, distributive 281 Module, exponent 69 Module, faithful 281 Module, finite 69 70 86 94 Module, free 52—56 58 60 61 302 Module, free, finite rank 59 Module, independent generators 96 97 Module, isomorphism 57 58 96 97 99 100 Module, submodule 69—71 70 95 99 100 252 253 255 256 258 Module, submodule, cyclic 99 Module, submodule, minimal 69 70 Module, torsion 68 97 98 Module, torsion, finite 68 71 93 Module, torsion, finitely generated 90 97 Module, type 94 Moebius, algebra 53—59 76 86 Moebius, algebra, basis 58 59 61—65 80 Moebius, algebra, isomorphism 58 Moebius, category 18 Moebius, function 12 29 33 35—45 47 51 55 58 59 65 66 08 69 71 72 81 82 84 86 91 96 101 120 Moebius, inversion 33—36 42 43 50 52 53 55 57 61 80 87 88 90 91 95 100 104 Monomial 145 297—299 301 Monomial, degree 297 Morphism 137—140 Multiplicative autoinorphisni 274 275 277 278 Multiplicative map 79 80 Multiplicatively closed set 234 283 241 246 Nest algebra 18 Normal series 82 Number theory 10 29 33 44 Omega ideal 209 211 212 232 233 238 239 Omega ideal, maximal 212—215 248 Omega ideal, prime 211 213 231 240 248—250 Omega ideal, prime, mitiimal 231 238—240 Open set 195 197 Orbit 102 Order complex 51 Order ideal 126 127 Ordinal number 181 183 184 228 229 Ordinal number, limit 181 183 184 228 Ore condition 284—287 292 295 296 p-socle 95 96 Partially ordered set 1 Partially ordered set, anti isomorphism 38 39 44 45 Partially ordered set, Automorphism 273 277—279 Partially ordered set, bounded 5 121 177 179 183—187 199 200 220 222 231 233 236—240 247 248 299 301 302 304 Partially ordered set, complement 63 64 83 84 Partially ordered set, connected 3 9 30 31 164 259 262 271 273 Partially ordered set, countable 14 16 17 224 265 270 Partially ordered set, direct product 39—44 69 117 120 Partially ordered set, doubly infinite 288—290 292 295 296 Partially ordered set, embedding 16 17 149 224 288 289 Partially ordered set, family of 141 148 161 Partially ordered set, finite 14 24 25 35 37 38 40 51 58 171 172 214 229 231 247 248 265 268 279 280 299—301 305 306 Partially ordered set, hereditary 144 146 149 151 153 157 Partially ordered set, ideal of a 34 Partially ordered set, ideal of a principal 34 Partially ordered set, infinite 23 213 223 231 236 247 248 281 Partially ordered set, interval 4 Partially ordered set, interval, length 5 11—13 113—116 293 294 Partially ordered set, irreducible element of 77—79 81 Partially ordered set, isomorphism 5 7—9 20 22 25 38 39 42 43 66 67 69 71 85 111 119 127 130 136 143—145 113 149 153 245 260 261 264 265 268 270—272 288 Partially ordered set, isomorphism, partial 264 265 269 Partially ordered set, isomorphism, potential 264 265 268 270 272 281 Partially ordered set, lower finite 18 173 281 Partially ordered set, maximal element 2 14 54 109 201 288 289 291 294 Partially ordered set, maximum clement 2 79 164 Partially ordered set, minimal element 2 6 199 204 220 221 225 227 287—289 293 295 Partially ordered set, minimum element 2 44 47 53 59 90 148 Partially ordered set, opposite 2 Partially ordered set, segment 4 Partially ordered set, subposet 2 3 5 7 14 19 31 32 45 49 54 77 79 81 102 107 100 120 164 245 264 287 288 Partially ordered set, subposet of periods 102 Partially ordered set, unbounded 5—7 31 117 120 178 183 187 220 221 224 233 240 245 248 300 Partition 6—9 20 21 71 72 90 92 102 103 107 132 134 144 145 148 159 213 244 271 Permutation 64 100 119 148 301 301 Permutation, similar 301—304 PID 68 69 71 86 93 96 97 99 Polya — de Bniijn Theorem 101 103 101 Polya's theorem 101 102 105 Polynomial 23 24 88—90 93 297—301 Polynomial identity 297—302 304 305 Polynomial identity, homogeneous 297 299 Polynomial sequence 133 134 140 Polynomial sequence, binomial type 133 134 140 Polynomial sequence, binomial type, multivariate 141 Polynomial sequence, Boolean type 134 Polynomial sequence, exponential 134 Polynomial sequence, multivariate 134 141 Polynomial, characteristic 23 Polynomial, chromatic 134 Polynomial, cyclotomic 88 Polynomial, degree 87 89 94 134 297—302 Polynomial, exponential Bell 159 Polynomial, homogeneous 297 301 Polynomial, homogeneous, linear 302 Polynomial, irreducible 87 94 100 Polynomial, minimal 87 93 Polynomial, monic 87 88 Polynomial, relatively prime 89 Polynomial, separable 87 Polynomial, standard 299 301 302 Power series 27 113 114 116 117 120 121 135 147 157 159 161 Power series, Eulerian 116 121 137 Power series, exponential 133 Pre-ordered set 1 4 9 18 173 240 Primitive root of unity 88 probability 93 Projection map 99 129 209 Projective space 161 Q-ideal 168 Q-radical 168 Radical 187 Radical, Brown — McCoy 171 Radical, Jacobson 168—170 193 200 202 272 Radical, lower nil 181—185 Radical, nil 173 174 176 179 184 185 222 230 231 238 239 Radical, prime 181 305 Radical, property 168 Radical, upper nil 181 184 185 Ramsey's theorem 5 Rank function 74 75 Reflexivity 1 Riemann zeta function 29 Ring, Artinian 305 306 Ring, automorphism 262 263 273 274 277 279—281 Ring, direct sum 57 280 Ring, embedding 19 285 Ring, finite 272 Ring, indecomposable 203 271 277 Ring, isomorphism 13 15 18 40 61 02 271 285 Ring, local 213 215 Ring, nil 180 Ring, nilpotent 180 184 Ring, nilpotent, element 176 177 180 181 184 Ring, Noetherian 272 305 Ring, of polynomials 53 133 146 157 207 Ring, of quotients 283—287 290 292 Ring, of quotients, isomorphism 284 Ring, of quotients, unit 12 13 23 36 59 289 201 295 Ring, prime 305 Ring, quotient 76 Ring, regular 241—243 Ring, regular, element 243 283—296 305 Ring, semi perfect 306 Ring, semilocal 213—215 248 249 Ring, semiprimary 305 Ring, semiprime 305 Ring, semisimple 305 306 Ring, subring 13 15 16 18 37 180 187 280 285 Ring, subring, nil 180 Ring, subring, nilpotent 180 Ring, theory 176 179 296 297 305 Ring, tic-tac-toe 18 Ring, unit 170 213 241—243 249 273 275 288 289 293 303 Ring, valuation 76 77 86 S-equivalcnce relation 20 32 Schur product 25 252 Scries-parallel networks 161 Semigroup 56 61 117 131 142 143 147 154 155 157 159 Semigroup, anti-isomorphism 159 Semigroup, direct sum 117 Semigroup, embedding 154 155 Semigroup, isomorphism 61 151 157 Semigroup, of multiplicative fund ions 150 157 159 Semigroup, of types 142 152—155 Semigroup, order compatible 142—144 i5l 152 154 155 Semigroup, subsemigroup 147 Set theory 272 Sheaf 18 Similarity class 93 101 Simpiicial complex 50 51 Simpiicial complex, dimension 50 Slone space 196—198 208 218 219 221 224 225 228—230 234 Special function 244—246 248 Spectrum 194—196 203 204 208 209 211 217 231—233 236 237 239 242 247—249 Spectrum, maximum 196—198 208 211 213 215 218 Spectrum, minimum 217—219 221 224 228 229 236—238 Strongly nilpotcnt 176—179 181 182 185 187 234 236 237 Structural matrix ring 16 18 System of distinct representatives 64 65 T nilpolent 306 topology 10 52 194—196 217 229 Topology, basis 195 196 Topology, discrete 10 Topology, induced 196 208 217 Topology, standard 10 202 Topology, Zariski 195 Totally ordered 2 17 Transitivity 1 TREE 117 120 TYPE 19 107 109—112 114 116 119 129 130 142—146 150—155 158—160 Type, cancelable 154 Type, one point 128 129 152—155 158 UFD 120 Ultrafilter 165—167 189 191 192 196—200 202 206 208—215 217—220 222 230 231 239 Ultrafilter, bounded 199 200 Ultrafilter, principal 202 215 230 231 238 Unimodular matroids 161 Unit map 17 Upper bound 163 164 Valuation 77 78 81 Valuation, extension 77 Vandermonde convolution 131 Variable 53 111 117 121 297—299 301 302 304 Variable, degree of 298 Variable, linear 298 Vector 67 145 148 Vector space 17 44 66—71 93—97 99 115 121 122 124—126 128—130 132 135—138 146 171 Vector space, basis 67 97 99 126 128 135 143 155 171 172 224 Vector space, dimension 66—69 71 93—97 115 116 136 171 Vector space, dimension, finite 115 136 171 Vector space, dimension, quotient 66 67 116 Vector space, dimension, Span 59—61 65 67 93 124 126 128 130 132 135 138 302 Vector space, dimension, subspace 66—68 71 115 122 124 129 136 138 140 Vector, component 148 Vector, linearly independent 53 60 61 60 67 78 99 126 224 302 Waring's formula 101 Zero divisor 240—242 283 287—289 Zero divisor, left 287 Zero divisor, right 287 288 291 Zero set 204 Zeta function 11 15 22—25 27 37 105 118 191 Zorm's Lemma 166 217
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