30,251 research outputs found
On scattered convex geometries
A convex geometry is a closure space satisfying the anti-exchange axiom. For
several types of algebraic convex geometries we describe when the collection of
closed sets is order scattered, in terms of obstructions to the semilattice of
compact elements. In particular, a semilattice , that does not
appear among minimal obstructions to order-scattered algebraic modular
lattices, plays a prominent role in convex geometries case. The connection to
topological scatteredness is established in convex geometries of relatively
convex sets.Comment: 25 pages, 1 figure, submitte
Statistical Lorentzian geometry and the closeness of Lorentzian manifolds
I introduce a family of closeness functions between causal Lorentzian
geometries of finite volume and arbitrary underlying topology. When points are
randomly scattered in a Lorentzian manifold, with uniform density according to
the volume element, some information on the topology and metric is encoded in
the partial order that the causal structure induces among those points; one can
then define closeness between Lorentzian geometries by comparing the sets of
probabilities they give for obtaining the same posets. If the density of points
is finite, one gets a pseudo-distance, which only compares the manifolds down
to a finite volume scale, as illustrated here by a fully worked out example of
two 2-dimensional manifolds of different topology; if the density is allowed to
become infinite, a true distance can be defined on the space of all Lorentzian
geometries. The introductory and concluding sections include some remarks on
the motivation for this definition and its applications to quantum gravity.Comment: Plain TeX, 19 pages + 3 figures, revised version for publication in
J.Math.Phys., significantly improved conten
Sublattices of lattices of convex subsets of vector spaces
For a left vector space V over a totally ordered division ring F, let Co(V)
denote the lattice of convex subsets of V. We prove that every lattice L can be
embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite
lower bounded, then V can be taken finite-dimensional, and L embeds into a
finite lower bounded lattice of the form ,
for some finite subset of . In particular, we obtain a new universal
class for finite lower bounded lattices
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
Finite convex geometries of circles
Let F be a finite set of circles in the plane. We point out that the usual
convex closure restricted to F yields a convex geometry, that is, a
combinatorial structure introduced by P. H Edelman in 1980 under the name
"anti-exchange closure system". We prove that if the circles are collinear and
they are arranged in a "concave way", then they determine a convex geometry of
convex dimension at most 2, and each finite convex geometry of convex dimension
at most 2 can be represented this way. The proof uses some recent results from
Lattice Theory, and some of the auxiliary statements on lattices or convex
geometries could be of separate interest. The paper is concluded with some open
problems.Comment: 22 pages, 7 figure
Greedy algorithms and poset matroids
We generalize the matroid-theoretic approach to greedy algorithms to the
setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli
(1998) [BNP]. We illustrate our result by providing a generalization of Kruskal
algorithm (which finds a minimum spanning subtree of a weighted graph) to
abstract simplicial complexes
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
- …