42,288 research outputs found
Reverse mathematics and properties of finite character
We study the reverse mathematics of the principle stating that, for every
property of finite character, every set has a maximal subset satisfying the
property. In the context of set theory, this variant of Tukey's lemma is
equivalent to the axiom of choice. We study its behavior in the context of
second-order arithmetic, where it applies to sets of natural numbers only, and
give a full characterization of its strength in terms of the quantifier
structure of the formula defining the property. We then study the interaction
between properties of finite character and finitary closure operators, and the
interaction between these properties and a class of nondeterministic closure
operators.Comment: This paper corresponds to section 4 of arXiv:1009.3242, "Reverse
mathematics and equivalents of the axiom of choice", which has been
abbreviated and divided into two pieces for publicatio
Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality
principles that are equivalent to the axiom of choice in set theory. Among
these are the principle asserting that every family of sets has a
-maximal subfamily with the finite intersection property and the
principle asserting that if is a property of finite character then every
set has a -maximal subset of which holds. We show that these
principles and their variations have a wide range of strengths in the context
of second-order arithmetic, from being equivalent to to being
weaker than and incomparable with . In
particular, we identify a choice principle that, modulo induction,
lies strictly below the atomic model theorem principle and
implies the omitting partial types principle
On --domains and star operations
Let be a star operation on an integral domain . Let \f(D) be the
set of all nonzero finitely generated fractional ideals of . Call a
--Pr\"ufer (respectively, --Pr\"ufer) domain if
(respectively, ) for all F\in
\f(D). We establish that --Pr\"ufer domains (and --Pr\"ufer
domains) for various star operations span a major portion of the known
generalizations of Pr\"{u}fer domains inside the class of --domains. We also
use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of
Ideals, Academic Press, New York--London, 1971], which gives several equivalent
conditions for an integral domain to be a Pr\"ufer domain, as a model, and we
show which statements of that theorem on Pr\"ufer domains can be generalized in
a natural way and proved for --Pr\"ufer domains, and which cannot be. We
also show that in a --Pr\"ufer domain, each pair of -invertible
-ideals admits a GCD in the set of -invertible -ideals,
obtaining a remarkable generalization of a property holding for the "classical"
class of Pr\"ufer --multiplication domains. We also link being --Pr\"ufer (or --Pr\"ufer) with the group Inv of -invertible -ideals (under -multiplication) being
lattice-ordered
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
- β¦