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Fraisse R. — Theory of Relations
Fraisse R. — Theory of Relations

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Название: Theory of Relations

Автор: Fraisse R.


Relation theory originates with Hausdorff (Mengenlehre 1914) and Sierpinski (Nombres transfinis, 1928) with the study of order types, specially among chains = total orders = linear orders. One of its first important problems was partially solved by Dushnik, Miller 1940 who, starting from the chain of reals, obtained an infinite strictly decreasing sequence of chains (of continuum power) with respect to embeddability. In 1948 I conjectured that every strictly decreasing sequence of denumerable chains is finite. This was affirmatively proved by Laver (1968), in the more general case of denumerable unions of scattered chains (ie: which do not embed the chain Q of rationals), by using the barrier and the better orderin gof Nash-Williams (1965 to 68).

Another important problem is the extension to posets of classical properties of chains. For instance one easily sees that a chain A is scattered if the chain of inclusion of its initial intervals is itself scattered (6.1.4). Let us again define a scattered poset A by the non-embedding of Q in A. We say that A is finitely free if every antichain restriction of A is finite (antichain = set of mutually incomparable elements of the base). In 1969 Bonnet and Pouzet proved that a poset A is finitely free and scattered iff the ordering of inclusion of initial intervals of A is scattered. In 1981 Pouzet proved the equivalence with the a priori stronger condition that A is topologically scattered: (see 6.7.4: a more general result is due to Mislove 1984): ie: every non-empty set of initial intervals contains an isolated elements for the simple convergence topology.

In chapter 9 we begin the general theory of relations, with thenotions of local isomorphism, free interpretability and free operator (9.1 to 9.3), which is the relationist version of a free logical formula. This is generalized by the back-and-forth notions in 10.10: the (k,p)-operator is the relationist version of the elementary formula (first order formula with equality).

Chapter 12 connects relation theory with permutations: theorem of the increasing number of orbits (Livingstone, Wagner in 12.4). Also in this chapter homogeneity is introduced, then more deeply studied in the Appendix written by Norbert Saucer.

Chapter 13 connects relation theory with finite permutation groups: the main notions and results are due to Frasnay. Also mention the extension to relations of adjacent elements, by Hodges, Lachlan, Shelah who by this mean give an exact calculus of the reduction threshold.

The book covers almost all present knowledge in Relation Theory, from origins (Hausdorff 1914, Sierpinski 1928) to classical results (Frasnay 1965, Laver 1968, Pouzet 1981) until recent important publications (Abraham, Bonnet 1999).

All results are exposed in axiomatic set theory. This allows us, for each statement, to specify if it is proved only from ZF axioms of choice, the continuum hypothesis or only the ultrafilter axiom or the axiom of dependent choice, for instance.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2000

Количество страниц: 456

Добавлена в каталог: 19.11.2008

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Предметный указатель
$dom_{T}(x)$      A.4.5
$H_{C}$ = homogeneous representative      A.2.6
$K_{n}$-free graph      A.1.4
$\alpha$-disjoint morphisms      11.6.3
$\alpha$-neighbor, $\alpha$-neighborhood      6.2.2
$\alpha$-neighbor, $\alpha$-neighborhood, neighborhood rank      6.2.4
$\Delta$-family      11.5
$\Delta$-family, solid or fragil family      11.5.1
(k,p)-connected element      11.8.2
0-ary relation (E,+) or (E,-)      1.7
1-extension      10.1.6
1-isomorphism, (1,p)-isomorphism      10.1.9
1-isomorphism, (k,p)-isomorphism      10.10.1
1-isomorphism, A-isomorphism      11.1.1
1-morphism, (l,p)-morphism      10.1.9
1-morphism, (l,p)-morphism, $\alpha$-morphism      10.4.1
1-morphism, (l,p)-morphism, $\alpha$-solid or fragil morphism      11.6
3-cycle, binary cycle      9.7
A-restriction      11.1.1
ABBOTT, Schur numbers      3.8.3
ABRAHAM, Hausdorff construction of scattered posets      9.9
Absorption      1.3.1 1.3.3
Accessible cardinal      2.8.9
Adherent element (topology)      6.6.2 6.6.4
Adherent, n.adherent permutation      12.3.4
Adjacence lemma, adjacent elements      13.11
Adjacent vertices (~)      A.1.4
Age      10.2.1
Age-indivisible      A.4
Age-indivisible, age-inexhaustible      A.3.1
Agrees with an interval-ultrafilter      11.7.2
AJDUKIEWICZ, couple      1.1.2
Aleph      1.6.3
Aleph, aleph rank      1.6.6
Aleph, sum, product, exponentiation      1.6.8
Almost chainable relation      10.9
Altered restriction (lemma)      9.1.3
Amalgam      A.1
Amalgamable set and amalgamable age      12.2
Amalgamable set and amalgamable age, freely amalgamable      A.2.2
Amalgamable set and amalgamable age, strongly amalgamable      A.2.1
Amalgamation lemma (posets)      1.7.3
Amalgamation lemma (posets), amalgamation function      A.1
Amalgamation lemma (posets), amalgamation theorem      12.2.1
Amalgamation lemma (posets), non-amalgamation for trees      2.11.6
Antichain      2.2.6
Antichain, poset of antichains $\mathcal{A}(X)$      6.7.3
Arity greater than another      9.3.3
Arity, group      13.3.1
Arity, operator      9.3
Arity, relation, multirelation      1.7
ARONSZAJN, tree      5.9.1
Assignable (multirelation)      9.3
ASSOUS, lexicographic rank of a barrier      7.1.3
Augmentation = reinforcement, augmented poset      2.9.1 6.5 9.9
Augmentation axiom      2.9.3
Automorphism      1.6 1.7.6
Automorphism lemma      9.1.2
Axiom of accessibility      2.8.9
Axiom of augmentation ( = reinforcement)      2.9.3
Axiom of Choice      1.1.8
Axiom of choice for finite sets      1.2.9
Axiom of constructibility      2.1.5
Axiom of denumerable subset      1.2.6
Axiom of Dependent Choice      1.8
Axiom of foundation      1.2 1.9.1
Axiom of foundation, equivalent scheme      1.2.7
Axiom of infinity      1.2.4
Axiom of maximal ideal      2.13
Axiom of ordering      2.4.4
Axiom of Suslin      2.2.7 5.8
Axiom of trichotomy (cardinals)      1.6.4
Axiom of well-ordering      1.6.2
Axioms of ZF      1.2.4
BACHMAN, natural sum and product      4.8.2
Back-and-forth notions      10.10
Bad barrier sequence      7.2.1
Bad sequence      4.2.1
BAIRE, condition      6.6.6
Barrier      7.1
Barrier, barrier partition theorem      7.1.4
Barrier, barrier sequence      7.2
Base |R|      1.6.1 1.7
Basic clopen set      6.6.1 6.6.4
BAUMERT, Schur numbers      3.8.3
Bendixson      6.7.1
BENEJAM, coherence lemma      2.4.3
BERCOV, orbits      12.4.2
BERGE, Ramsey number and binomial coefficient      3.8.5
BERNAYS, axiom of foundation      1.9.1
BERNSTEIN, Bernstein — Schroeder for equimorphisms      5.1.2
BERNSTEIN, equipotence theorem      1.1.4 1.1.5 1.6.8
BERNSTEIN, separation lemma      2.3.3
Better partial ordering      7.6
Better partial ordering, better partial ordering (w. r. to barriers)      7.7
Betwenness = intermediacy relation      9.2.1
Bichain      13.5
Bichain, Q-bichain      13.7.1
Bijection = bijective function      1.1.2
Binary relation      1.6
Binary relation, binary cycle      9.7
Bipartite graph = bivalent tableau      8.4
Birelation, trirelation, quadrirelation      1.7
BIRKHOFF, finitely generated initial intervals      4.1.1
Bivalent tableau      8.4
BLASS, model without ultrafilters      2.3.5
BLASS, model without ultrafilters, axiomatic      2.8.10
BOEROECKY, age-inexhaustible system      A.3.3
BONNET, covering by indecomposable chains      7.5.5
BONNET, finitely free posets      4.7.2
BONNET, Hausdorff construction of scattered posets      9.9
BONNET, incomparable ideals      4.7.3 4.11.2
BONNET, number of initial intervals      6.7.4
BONNET, partition in slices      2.10.2
BONNET, scattered posets      6.5.1—6.5.3
BONNET, set $\mathcal{J}$ of initial intervals      2.9.4
BONNET, Szpilrajn chains      8.6.8
Boolean prime ideal axiom = ultrafilter axiom      2.13
Bound of a universal class      5.10.1
Bound of an age = bound of a relation      13.1.1
Bound of an initial interval (in a poset)      4.10
Bounded profile of an almost chainable relation      10.9.7
Bracelet (inextensivity)      8.5.1
CALAIS, homogeneity      12.2.2
CALAIS, homogeneity, pseudo-amalgamation theorem      12.6
CAMERON, homogeneous structure      A.2.8
CAMERON, Jordan's hypothesis      12.3.3
CAMERON, set-transitive group theorem      13.8
Canonical extension of an operator ($\mathcal{P}^{=}$)      9.3.5
Canonical partition      A.5.3
CANTOR, lemma, theorem      1.1.6
CANTOR, normal form or decomposition      1.3.5
Cantorian theorem for posets      5.2.3
Cardinal = cardinality (Card a) of a set      1.2.3 1.4.4
Cardinal = cardinality (Card a) of a set, of a relation, multirelation      1.7
Cardinal sum ($a \biguplus b$), product ($a \times b$), exponentiation ($^{a}b$)      1.4.5
Cardinal sum ($a \biguplus b$), product ($a \times b$), exponentiation ($^{a}b$), between alephs      1.6.8
Cartesian product ($a \times b$)      1.1.2
Cartesian product ($a \times b$), generalized      1.1.9
Center of a solid family      11.5.1
Chain = total ordering      1.6.1
Chain = total ordering, chain Q of rationals, chain R of reals      2.1.1
Chain = total ordering, scattered chain      6.1
Chain associated with a tree      2.11.2
Chain condition      A.4.5
Chain meeting every height      4.6.1
Chain of initial intervals      6.1.4
Chainability theorem      13.3.3
Chainability theorem, chainable relation      9.5
CHANG, 1-extension      10.1.6
CHAUNIER, computation of posets      2.14
CHERLIN, homogeneous directed graphs      A.1.7
Choice axiom, choice set and function      1.1.8
Choice for finite sets      1.2.9
CHUNG, Ramsey numbers      3.1.6
CLARK, (G, A)-chain      13.10.3
Classification of ages      11.4.1
Classification of cardinals      1.6.9
Clopen set      6.6 6.7.1
Closed under adherence (group)      12.3.4
Closed under embeddability      10.2
Co-initial restriction or subset      2.7.2
Co-initial restriction or subset, co-initial set of reals      2.1.1
Co-initiality      2.7.3
Cofinal height (Cofh)      2.12.3
Cofinal restriction of a net      2.13.2
Cofinal restriction or subset      2.7.2
Cofinal restriction or subset, cofinal set of reals      2.1.1
Cofinality (Cof)      2.7.3
COHEN, axiom of choice      1.2.4
COHEN, continuum hypothesis      1.5.4
COHEN, generalized continuum hypothesis      1.9.3
Coherence lemma      2.4.1
Coherence lemma, variant      2.4.3
COLOR      3.1.1
Color, good coloration      3.8.1
Column (in a tableau)      8.4
Commutative (= natural) sum $a \oplus b$      6
Commutative (= natural) sum $a \oplus b$, product $a \oplus b$      4.8.2
Compact (topology)      6.6.3 6.6.4
Comparison $\geq_{T}$      A.4.5
Compatibility modulo a group      13.3.1
Compatibility theorem (recollement)      13.3.2
Compatibility threshold      13.10.2
Compatible multirelations, relations      1.7.2
Complete graph $K_{n}$ on n vertices      A.1.4
Complete relational system      A.2.4
Completion of a successor barrier      7.3.2
Completion of a successor barrier sequence      7.3.4
Composition ($g \circ f$)      1.1.4
Conjunction, posets      4.9
Conjunction, relations (R $\wedge$ 5)      1.7.4
Consecutivity relation      9.8
Constant relation      9.4
Constructibility axiom      2.1.5
Continuum      1.5.1
Continuum Hypothesis      1.5.4
Continuum hypothesis, connection with $^{\omega}2=\omega_{1}$      1.6.7
Contracted group      13.5
Convergent sequence      6.6.2
Converse $R^{-}$      1.7.5
Copy of a relational system      A.5.1
COROMINAS, well quasi-ordering of trees      7.5.4
Countable dense set      5.3.4
Countable set, countable axiom of choice      1.2.5
Couple = ordered pair      1.1.2
Covering by doublets      6.4.5
Covering by right (or left) indecomposable chains      6.4.3
Criterion for a prehomogeneous relation      12.9
Criterion for a rich relation      11.4
CULBERSON, computation of posets      2.14
Cut      2.1.1 2.6.4 2.6.5
Cyclic order      A.2.7
Cyclic ternary relation associated with a chain      9.2.1
DAS, computation of posets      2.14
DAVIS, definition by recursion      1.2.10
DE JONGH, maximal augmented chain theorem      4.11.2
Decomposable chain      6.3
Decomposable ordinal      1.3.6
Decomposable ordinal sequence      7.6
Decomposition of a chain (Hausdorff)      6.2.1
Decomposition of a scattered chain      6.2.5 6.2.6
DEDEKIND, finite set      1.1.4
DEDEKIND, theorem      2.1.2
DEDEKIND, theorem, generalized in      2.6.4
Definition by recursion      1.2.10
Degree of a universal class      13.12.2
DEMBOWSKI, 2-set-transitive group      9.7.4
Denis DEVLIN, partitionning $Q^{n}$      5.12.2 A.5.5
Dense chain      5.3.1
Dense chain, $\alpha$-dense chain      5.7
Dense set in a chain      2.6.6
Dense set in a chain, dense set of reals      2.1.1
Denumerable partition of the continuum      1.5.3
Denumerable set      1.2.5
Denumerable subset axiom      1.2.6
Denumerably Szpilrajn chain      8.6.4
Dependent choice (axiom)      1.8
Dihedral permutation group $D_{m}$      9.2.2 13.6.2
Dihedral quaternary relation associated with a chain      9.2.2
Dilated group      13.4
DILWORTH, finitely free poset      4.14.1
DILWORTH, finitely free poset, Cantorian theorem for posets      5.2.3
Dimension (poset)      4.9
Direct product $A \times B$ of chains      4.9.3
Direct product $A \times B$ of posets      4.8
Directed poset = net      2.13
Directed poset = net, directed under embeddability      10.2
Disjunction $R \vee S$      1.7.4
Domain (Dom)      1.1.2
Doublet      6.4.4
du Bois-Reymond      5.11.1 5.11.2
DUSHNIK, partition theorem      3.3.2 3.3.3
DUSHNIK, partition theorem, decreasing sequence      5.5.2 9.5.5
DUSHNIK, partition theorem, dimension      4.9
EDGE      3.1.1 A.1.4
Edge, up edge, down edge      A.5.2
Edge-indivisible graph      A.5.1
EL-ZAHAR, condition for indivisibility      A.4.5
EL-ZAHAR, divisibility and squares      A.4.10
EL-ZAHAR, indivisible graph      A.4.11
EL-ZAHAR, property P      A.4.1
EL-ZAHAR, strongly amalgamable age      A.3.5
EL-ZAHAR, weak indivisibility      A.4.3
Elementary equivalence      10.10.1
Elementary extension      10.10.3
ELLENTUCK, Ramsey set      3.7.1
Embedding, A-embedding      11.1.1
Embedding, relations      5.1.1
embedding, sequences      4.1.2
Empty base      1.7 10.10.4
empty function      1.7.6 10.4.1 10.10.1
Epsilon ordinal      1.3.3
Equimorphic, equimorphism ($\backsimeq$)      5.1.1
Equipotence      1.1.4
Equivalence $\overset{n}{\sim}$      A.3.3
Equivalence relation, (1,p)-equivalence      10.1.10
Equivalence relation, (k,p)-equivalence      10.10.1
Equivalence relation, class      1.6.1
Equivalent tableaux      8.5.3
ERDOES, with Dushnik, Miller      3.3.3
ERDOES, with Dushnik, Miller, partition lemma      3.3.4
ERDOES, with Dushnik, Miller, partition lemma, edge-indivisible graph      A.5.2
ERDOES, with Dushnik, Miller, partition lemma, partition theorem      3.3.5
ERDOES, with Dushnik, Miller, partition lemma, Ramsey numbers      3.8.1
ERDOES, with Dushnik, Miller, partition lemma, Ramsey set      3.7.1
Essentially equivalent relational system      A.1.7
Existence criterion for a prehomogeneous relation      12.9
Existence criterion for a rich or saturated relation      11.4
Existentially closed = maximalist relation      11.2.6
EXOO, Ramsey numbers      3.1.6
Exponentiation (cardinal), notation $^{a}b$ between cardinals      1.4.5
Exponentiation (cardinal), notation $^{a}b$ between sets      1.1.7
Exponentiation between ordinals, $\alpha^{\beta}$      1.3.3
Exponentiation between ordinals, Hessenberg exponentiation      9.9.2
Extendomorphic h-tuples      9.8.3
Extension of a function      1.1.4
Extension of a function, extension of a relation or multirelation      1.7.1
Extension of a function, extension of an operator      9.3.5
Extensive subset      12.5
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