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Monk J.D. (ed.) — Handbook of Boolean Algebras, Vol. 1
Monk J.D. (ed.) — Handbook of Boolean Algebras, Vol. 1



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Название: Handbook of Boolean Algebras, Vol. 1

Автор: Monk J.D. (ed.)

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second

Год издания: 1989

Количество страниц: 344

Добавлена в каталог: 09.06.2009

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Предметный указатель
$(\kappa, \infty)$-distributive law      213 228
$(\kappa, \lambda)$-distributive law      213 217ff
$(\kappa, \lambda, \mu)$-distributive      223f
$(\kappa, \lambda, \mu)$-nowhere distributive      223ff
$Fr\kappa$      132ff 247
$F_{\sigma}$-set in a topological space      106 176
$P(\omega)/fin$      47 78ff 95 176 215
$\Delta$-system      136 166
$\kappa$-algebra of sets      21 214
$\kappa$-antichain condition      42
$\kappa$-Aronszajn tree      228
$\kappa$-chain condition      41 140ff
$\kappa$-closed cardinal      146
$\kappa$-closed partial order      215
$\kappa$-complete BA      20 104 173 175
$\kappa$-complete homomorphism      66 77
$\kappa$-complete ideal      75
$\kappa$-complete subalgebra      50 140
$\kappa$-completely generated      141ff
$\kappa$-refinement of a partition      226
$\kappa$-representable      182 221f
$\kappa$-Souslin algebra      228ff
$\kappa$-Souslin tree      228ff 252
$\kappa$-weak product      86
$\omega$-saturated      293f
$\omega_{1}$-Saturated      82 85
$\omega_{1}$-universal      81
$\phi$-homogeneous      198
$\pi A$      63
$\pi$-character      153ff
$\pi$-weight      54
$\sigma$-additive measure      46 85 233
$\sigma$-algebra of projections      24
$\sigma$-algebra of sets      21 51 189
$\sigma$-complete BA      20 40 46 61 88 103 163 175 189 211 see
$\sigma$-complete homomorphism      66
$\sigma$-complete ideal      75 182 233
$\sigma$-complete subalgebra      50
$\sigma$-representable BA      182 221
$\tau A$      199ff 204ff
A-valued L-structure      193
Absolute of X      125
Absorption law      8
Additive normal form of an element      51
Algebra of formulas      11 12 133f see
Algebra of Lebesgue measurable sets      21 234
Algebra of sets      5 8f 10 30 214
Algebraic Theory, $<x_{i}:i\in I>$, $<\{x_{i}:i\in I\}>$      51
Algebraic Theory, $A(x_{1},\ldots, x_{n})$, $<A\cup\{x_{1},\ldots, x_{n}\}>$      53
Algebraic Theory, $A^{n}$, $^{n}A$      86
Algebraic Theory, $A_{1}\times\cdots\times A_{n}$, product of $A_{1},\ldots, A_{n}$      86
Algebraic Theory, $Fn(I, J, \lambda)$, partial functions from I into J      55
Algebraic Theory, $pr_{i}$, projection map from a product to a factor      86
Algebraic Theory, $u_{p}$, $\{q\in P:q\leq p\}$      55
Algebraic Theory, $\bar{A}$, completion of A      59
Algebraic Theory, $\equiv$, congruence relation associated with an ideal      76
Algebraic Theory, $\mathbf{BA}$, class of all BAs      49
Algebraic Theory, $\omega^{*}$, $\beta\omega\backslash\omega$      78
Algebraic Theory, $\pi B$, $min\{|X|: X\subseteq B dense\ in B\}$      54
Algebraic Theory, $\pi(x)$, congruence class of x      74
Algebraic Theory, $\prod^{<\kappa}_{i\in I}A_{i}$, $\kappa$-weak product      87
Algebraic Theory, $\prod^{w}_{i\in I}A_{i}$, weak product      87
Algebraic Theory, $\prod_{i\in I}A_{i}$, cartesian product of the algebras $A_{i}$      86
Algebraic Theory, $\prod_{i\in I}A_{i}/D$, ultraproduct of the algebras $A_{i}$      92
Algebraic Theory, $^{I}A$, product of A repeated I times      86
Algebraic Theory, (+1)x, x      51
Algebraic Theory, (-1)x, -x      51
Algebraic Theory, -F, ideal dual to I      75
Algebraic Theory, -I, filter dual to I      75
Algebraic Theory, <X>, subalgebra generated by X      51
Algebraic Theory, A(x), $&lt;A\cup\{x\}&gt;$      53
Algebraic Theory, A/F, quotient algebra by a filter F      76
Algebraic Theory, A/I, quotient algebra by an ideal I      76
Algebraic Theory, A/~, set of equivalence classes under ~      75
Almost disjoint sets      80 157 278
Amalgamated free products      127 157 168 173 185f
Antichains      42 64 253 see
Antichains in a partially ordered set      64 252
Antichains in a tree      228
Antichains in subalgebras of interval algebras      239 252ff
Argyros proposition on independence      138
Argyros, S.      138 144 309
Aronszajn tree      213 228ff 231
Associated (measure algebras and measure spaces)      233
Associativity law      7 22
atom      5 29 7 103 172
Atomic BA      29 64
Atomless BA      29 64 82 103 134
Attainment of cellularity      38 41 166
Automorphism      66 135 139 164 190 207 263ff
Automorphisms of complete BAs      173
Axiom of choice (and Boolean prime ideal theorem)      33 35
Axioms      7
B(T)      12 28 38 133 see "Lindenbaum
Back and forth property      72
Baire $\sigma$-algebra      21 28
Baire property      21
Baire's theorem      182 223
Balcar — Franek theorem      38 129 140 173 175 190 196ff
Balcar — Vojtas theorem      5 7 38 43 44ff 81 202
Balcar, B.      2 132 139
Base space of a sheaf      117
Basically disconnected space      103
Baumgartner, J.      252 309
Behrends, E.      2
Bidual      100
Birkhoff's theorem on varieties      49
Bonnet, R.      2
Boole, G.      3 309
Boolean algebra (definition)      7
Boolean algebra of projections      5 7 20 23
Boolean equivalence relations      93 107 109
Boolean order      244 272
Boolean pair      287 299
Boolean partial order on A      15
Boolean prime ideal theorem      33 36 154
Boolean ring      5 18 19
Boolean space      93 96
Boolean truth value      193
Boolean valued models      3 58 190
Borel algebra of X      21 24 182
Borel set      21
BPI      33 see
Branch of a tree      228
Brenner — Monk proposition (on chains in a tree algebra)      269
Brenner, G.      241 255 267 269f 309
Burris, S.      287 300 309
C*-embedded      177
CA      41ff 64 166 199ff see "Chain
Canonical generators of a tree algebra      255
Canonical homeomorphism      107
Canonical homomorphism w.r.t. a congruence relation      75
Canonical neighborhood (in a sheaf)      117
Canonical partial order on A      15
Cantor discontinuum      97
Cantor space      96 97 101 104 130 133 162 184
Cantor — Bendixson invariants      239 241 271 273 275 288 293
Cardinal function      38 198ff
Cardinal sequence of a BA      239 271 277
Cardinality of a BA      8 10 40 53 177 199
Cartesian product      39 49 85ff 107
Categories of BAs      87 106 159 175
Category of $\kappa$-complete BAs      185ff
Cellularity of a BA      5 38 41 166 199 see "Chain
Cellularity of a partial order      64
Cellularity of a topological space      41f
ch      37 252 see
Chain conditions      127 140 164 see "Cellularity"
Chain conditions in free BAs      137
Chain conditions in subalgebras of interval algebras      239 252ff
Chain conditions in tree algebras      269
Chains in BAs      42 46 137 241ff
Chang, C.C.      287 309
Characteristic homomorphism      32
Characterization of a free BA      131
Characterization of amalgamated free products      169
Characterization of free products      160
Clopen algebra      10 99ff
Closed under an equivalence relation      109
Closure in a space      25
Closure properties of interval algebras      239 246
Closure properties of tree algebras      239 265
Co-zero set      177
Cofinite      10
Cohen, P.J.      4
Collapsing algebra      175 190ff 212 225ff 236
Comer, S.      300
Comfort, W.W.      208
Commutative square      185
Commutativity law      8
Compact element of a lattice      84
Compact Hausdorff space      37 95
Compactification      93 111 112
Comparable elements of a BA      62
Compatible elements of a partial order      55
Complement      8 16
Complementation law      8
Complemented lattice      7 16
Complete algebra of projections      24
Complete algebra of sets      21 50f
Complete BA      20 27 60 103 173 175 190 228ff see
Complete generation      199 204f 228f
Complete homomorphism      66
Complete ideal      75
Complete linear order      243
Complete subalgebra      50
Completely distributive      213ff
Completely generated subalgebra      141 191 199 205 229
Completely separated      177
Completeness theorem for first order logic      38
Completeness Theorem for propositional logic      38
Completion of a BA      47 59 84 102 181
Completion of a partial order      47 54f 58 84 181
Completion of a theory      37 298
Cone in $\mathbf{C}_{\kappa}$      185
Congruence relation on a BA      74
Congruence relation on a L-structure      301
Connected space      97
Consequences of a theory      298
Continuous map      93 106
Continuum Hypothesis      37 62 82 231 252 see
Contravariant functor      106
Convex equivalence relation on a linear order      247
Convex subset of a linear order      246
Countable BAs      8 72 104 247 267
Countable chain condition      41 46 65 137 157 206 235
Countable separation property      79 139 172ff 189 206 211 249 263
Countable-cocountable algebra      211
Countably completely generated BA      173 190
Cylindric algebras      v 300
de Groot, J.      145
De Morgan's laws      10 11 16ff 22 36
Decidability      285
Decidable theories      107 287
Defines (formula defines a subset)      291
Dense (for a subset of a BA)      47 54 153
Dense set in a space      106
density      54ff 199
Devlin, K.      231 309
Directed family      52
Disjoint (ideal and subalgebra)      250
Disjoint elements      39
Disjoint refinement property      5 38 43ff 46 202
Disjoint union space      112
Disjunctive normal form      51
Distributive lattice      7 16
Distributive laws      8 15 18 22 173 212ff see
Dual algebra      100
Dual kernel of a homomorphism      76
Dual of a continuous map      107
Dual of a homomorphism      107
Dual properties      93 102
Dual space      190
Dual statement      13
Duality      107
Duality of homomorphisms, continuous maps      108ff
Duality principle      5 13
Dunford, N.      23 309
Dwinger, Ph.      115 309
Efimov, B.      309
Elementarily equivalent (in $L_{\infty\omega}$)      84
Elementary Arithmetic, $A\cong B$, A is isomorphic to B      9
Elementary Arithmetic, $A\lceil a$, relative algebra of A with respect to a      39
Elementary Arithmetic, $A\times B$, cartesian product of A and B      39
Elementary Arithmetic, $A^{+}$, set of non-zero elements of A      28
Elementary Arithmetic, $c_{A}a$, $c(A\lceil a)$      42
Elementary Arithmetic, $p_{a}$, projection map from A onto $A\lceil a$      39
Elementary Arithmetic, $\cdot$, basic operation, corresponding to intersection      7
Elementary Arithmetic, $\infty$, element adjoined to a linear order L      10
Elementary Arithmetic, $\leq$, ordering on a BA: $x\leq y$ iff $x+y = y$      14
Elementary Arithmetic, $\prod M$, greatest lower bound of M      20
Elementary Arithmetic, $\prod^{A}M$, $\prod M$ in A      20
Elementary Arithmetic, $\prod_{i\in I}m_{i}$, $\prod\{m_{i}:i\in I\}$      20
Elementary Arithmetic, $\sum M$, least upper bound of M      20
Elementary Arithmetic, $\sum^{A}M$, $\sum M$ in A      20
Elementary Arithmetic, $\sum_{i\in I}m_{i}$, $\sum\{m_{i}:i\in I\}$      20
Elementary Arithmetic, $\triangle$, symmetric difference      18
Elementary Arithmetic, +, basic operation, corresponding to union      7
Elementary Arithmetic, -, basic operation, corresponding to complementation      7
Elementary Arithmetic, 0, basic constant, corresponding to the empty set      7
Elementary Arithmetic, 1, basic constant, corresponding to the universal set      7
Elementary Arithmetic, 2, two-element BA      9
Elementary Arithmetic, At A, set of all atoms of A      29
Elementary Arithmetic, B(T), Lindenbaum — Tarski algebra of T      12
Elementary Arithmetic, Bai X, Baire algebra of X      21
Elementary Arithmetic, Bor X, algebra of Borel subsets of X      21
Elementary Arithmetic, c A, cellularity of A (supremum of |X|, X pairwise disjoint)      41
Elementary Arithmetic, c X, cellularity of the space X      42
Elementary Arithmetic, cl a, closure of a      25
Elementary Arithmetic, Clop X, BA of clopen subsets of X      10
Elementary Arithmetic, f[A], image of A under f      29
Elementary Arithmetic, glb M, greatest lower bound of M      14
Elementary Arithmetic, int a, interior of a      25
Elementary Arithmetic, Intalg L, interval algebra of L      11
Elementary Arithmetic, Leb $\mathbf{R}$, algebra of Lebesgue-measurable subsets of $\mathbf{R}$      21
Elementary Arithmetic, lub M, least upper bound of M      14
Elementary Arithmetic, P(X), power set of X      9
Elementary Arithmetic, r a, int cl a, the regularization of a      25
Elementary Arithmetic, RO(X), regular open algebra of X      25
Elementary Arithmetic, s(x), $\{p\in Ult A:x\in p\}$, the Stone map      33
Elementary Arithmetic, sat A, saturation of A (least $\kappa&gt;|X|$ for all X pairwise disjoint)      41
Elementary Arithmetic, Ult A, set of all ultrafilters on A      33
Elementary Arithmetic, x<y, $x\leq y$ and $x\neq y$      28
Elementary Arithmetic, [x, y), $\{z\in L:x\leq z&lt;y\}$ for a linear order L      10
Elementary derivative (ith)      288
Elementary equivalence      285ff
Elementary invariants of a BA      285ff
Elementary product      51 256
Elementary type      85 299
Elimination of quantifiers      85
Embedding      29
Endomorphisms of BAs      66
Engelking, R.      95 114 125 309
Epimorphism      29
Equation holds in a BA      34 49
Equivalence of compactifications      116
Equivalent (for $L_{\infty\omega}$)      73
Erdoes — Rado theorem      172
Erdoes — Tarski example      166
Erdoes — Tarski theorem      5 7 41ff 158 166 202f
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