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

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

Join-semidistributive lattices of relatively convex sets

We give two sufficient conditions for the lattice Co(R^n,X) of relatively convex sets of n-dimensional real space R^n 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(R^n,X), for a suitable finite subset X of R^n.Comment: 11 pages, first presented on AAA-65 in Potsdam, March 200

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

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

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

Algebraic convex geometries revisited

Representation of convex geometry as an appropriate join of compatible orderings of the base set can be achieved, when closure operator of convex geometry is algebraic, or finitary. This bears to the finite case proved by P. Edelman and R. Jamison to the greater extent than was thought earlie

Representation of algebraic convex geometries

Convex geometry is a set system generated by the closure operator with the antiexchange axiom. These systems model the concept of convexity in various settings. They are also closely connected to anti-matroids, which are set systems with the property of accessibility. In particular, the latter were used in modelling the states of human learners and found practical applications in designing the automatic tutoring systems. In current work we develop the theoretical foundations of infinite convex geometries in case their closure operator satisfies the finitary property: closure of any subset is a union of closures of its finite subsets. In such case, the convex geometry is called algebraic