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Theory of Relations, Volume 145 (Hardcover, Rev ed.)
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Theory of Relations, Volume 145 (Hardcover, Rev ed.)
Series: Studies in Logic and the Foundations of Mathematics
Expected to ship within 12 - 19 working days
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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 the
notions 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.
General
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