252 research outputs found
A note on blockers in posets
The blocker of an antichain in a finite poset is the set of
elements minimal with the property of having with each member of a common
predecessor. The following is done:
1. The posets for which for all antichains are characterized.
2. The blocker of a symmetric antichain in the partition lattice is
characterized.
3. Connections with the question of finding minimal size blocking sets for
certain set families are discussed
Chains, Antichains, and Complements in Infinite Partition Lattices
We consider the partition lattice on any set of transfinite
cardinality and properties of whose analogues do not hold
for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the
cardinality of any maximal well-ordered chain is always exactly ; (II)
there are maximal chains in of cardinality ; (III) if,
for every cardinal , we have , there
exists a maximal chain of cardinality (but ) in
; (IV) every non-trivial maximal antichain in has
cardinality between and , and these bounds are realized.
Moreover we can construct maximal antichains of cardinality for any ; (V) all cardinals of the form
with occur as the number of
complements to some partition , and only these
cardinalities appear. Moreover, we give a direct formula for the number of
complements to a given partition; (VI) Under the Generalized Continuum
Hypothesis, the cardinalities of maximal chains, maximal antichains, and
numbers of complements are fully determined, and we provide a complete
characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201
Short antichains in root systems, semi-Catalan arrangements, and B-stable subspaces
Let \be be a Borel subalgebra of a complex simple Lie algebra \g. An
ideal of \be is called ad-nilpotent, if it is contained in [\be,\be]. The
generators of an ad-nilpotent ideal give rise to an antichain in the poset of
positive roots, and the whole theory can be expressed in a combinatorial
fashion, in terms of antichains. The aim of this paper is to present a
refinement of the enumerative theory of ad-nilpotent ideals for the case in
which \g has roots of different length. An antichain is called short, if it
consists of short roots. We obtain, for short antichains, analogues of all
results known for the usual antichains.Comment: LaTeX2e, 20 page
Combinatorial symbolic powers
Symbolic powers are studied in the combinatorial context of monomial ideals.
When the ideals are generated by quadratic squarefree monomials, the generators
of the symbolic powers are obstructions to vertex covering in the associated
graph and its blowups. As a result, perfect graphs play an important role in
the theory, dual to the role played by perfect graphs in the theory of secants
of monomial ideals. We use Gr\"obner degenerations as a tool to reduce
questions about symbolic powers of arbitrary ideals to the monomial case. Among
the applications are a new, unified approach to the Gr\"obner bases of symbolic
powers of determinantal and Pfaffian ideals.Comment: 29 pages, 3 figures, Positive characteristic results incorporated
into main body of pape
An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games
An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in . This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u
On structures in hypergraphs of models of a theory
We define and study structural properties of hypergraphs of models of a
theory including lattice ones. Characterizations for the lattice properties of
hypergraphs of models of a theory, as well as for structures on sets of
isomorphism types of models of a theory, are given
A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees
Building on early work by Stevo Todorcevic, we describe a theory of
stationary subtrees of trees of successor-cardinal height. We define the
diagonal union of subsets of a tree, as well as normal ideals on a tree, and we
characterize arbitrary subsets of a non-special tree as being either stationary
or non-stationary.
We then use this theory to prove the following partition relation for trees:
Main Theorem: Let be any infinite regular cardinal, let be any
ordinal such that , and let be any natural
number. Then
This is a generalization to trees of the Balanced
Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above
to the cardinal , the simplest example of a
non--special tree.
As a corollary, we obtain a general result for partially ordered sets:
Theorem: Let be any infinite regular cardinal, let be any
ordinal such that , and let be any natural
number. Let be a partially ordered set such that . Then Comment: Submitted to Acta Mathematica Hungaric
The saturation spectrum for antichains of subsets
Extending a classical theorem of Sperner, we characterize the integers
such that there exists a maximal antichain of size in the Boolean lattice
, that is, the power set of , ordered by inclusion.
As an important ingredient in the proof, we initiate the study of an extension
of the Kruskal-Katona theorem which is of independent interest. For given
positive integers and , we ask which integers have the property that
there exists a family of -sets with
such that the shadow of has size , where the shadow of
is the collection of -sets that are contained in at least
one member of . We provide a complete answer for .
Moreover, we prove that the largest integer which is not the shadow size of any
family of -sets is .Comment: This is a merger of arXiv:2106.02226v2 with arXiv:2106.0223
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