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 $\Omega(\eta)$, 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