60 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
Stasheff polytope as a sublattice of permutohedron
An assosiahedron Kn, known also as Stasheff polytope, is a multifaceted
combinatorial object, which, in particular, can be realized as a convex
hull of certain points in Rn, forming (n − 1)-dimensional polytope1.
A permutahedron Pn is a polytope of dimension (n−1) in Rn with vertices
forming various permutations of n-element set. There exists well-known orderings
of vertices of Pn and Kn that make these objects into lattices: the first
known as permutation lattices, and the latter as Tamari lattices. We provide a
new proof to the statement that the vertices of Kn can be naturally associated
with particular vertices of Pn in such a way that the corresponding lattice
operations are preserved. In lattices terms, Tamari lattices are sublattices
of permutation lattices. The fact was established in 1997 in paper by Bjorner
and Wachs, but escaped the attention of lattice theorists. Our approach to the
proof is based on presentation of points of an associahedron Kn via so-called
bracketing functions. The new fact that we establish is that the embedding
preserves the height of element
Notes on the description of join-distributive lattices by permutations
Let L be a join-distributive lattice with length n and width(Ji L) \leq k.
There are two ways to describe L by k-1 permutations acting on an n-element
set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a
recent lattice theoretical way of the second author. We prove that these two
approaches are equivalent. Also, we characterize join-distributive lattices by
trajectories.Comment: 8 pages, 1 figur
A class of infinite convex geometries
Various characterizations of finite convex geometries are well known. This
note provides similar characterizations for possibly infinite convex geometries
whose lattice of closed sets is strongly coatomic and lower continuous. Some
classes of examples of such convex geometries are given.Comment: 10 page
Realization of abstract convex geometries by point configurations, Part 1
The Edelman-Jamison problem is to characterize those abstract convex
geometries that are representable by a set of points in the plane. We show that
some natural modification of the Edelman-Jamison problem is equivalent to the
well known NP-hard order type problem
Representing finite convex geometries by relatively convex sets
A closure system with the anti-exchange axiom is called a convex
geometry. One geometry is called a sub-geometry of the other if its closed sets
form a sublattice in the lattice of closed sets of the other. We prove that convex
geometries of relatively convex sets in n-dimensional vector space and their
nite sub-geometries satisfy the n-Carousel Rule, which is the strengthening
of the n-Carath eodory property. We also nd another property, that is similar
to the simplex partition property and does not follow from 2-Carusel Rule,
which holds in sub-geometries of 2-dimensional geometries of relatively convex
sets
Join-semidistributive lattices of relatively convex sets
We give two sufficient conditions for the lattice Co(Rn,X) of rel-
atively convex sets of Rn to be join-semidistributive, where X is a finite union
of segments. We also prove that every finite lower bounded lattice can be
embedded into Co(Rn,X), for a suitable finite subset X of R
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