Clustering on antimatroids and convex geometries

Several Aspects of Antimatroids and Convex Geometries Master's Thesis

Convexity is important in several elds, and we have some theories on it. In this thesis, we discuss a kind of combinatorial convexity, in particular, antimatroids and convex geometries. An antimatroid is a combinatorial abstraction of convexity. It has some di erent origins; by Dilworth in lattice theory, by Edelman and Jamison in the notions of convexity, by Korte{Lov asz who were motivated by...

Antimatroids and Convex Geometries – The fundamentals

Mathematical Morphology and Poset Geometry

The aim of this paper is to characterize morphological convex geometries (resp., antimatroids). We define these two structures by using closure operators, and kernel operators. We show that these convex geometries are equivalent to poset geometries. 2000 Mathematics Subject Classification. 37F20, 06A07.

A greedy algorithm for convex geometries

Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as dual of antimatroids. We consider functions defined on the sets of the extreme points of a convex geometry. Faigle– Kern (1996) presented a greedy algorithm to linear programming problems for shellings of posets, and Krüger (2000) introduced b-submodular functions and proved that Faigle–Kern’s algor...

Quasi-concave functions on antimatroids

In this paper we consider quasi-concave set functions defined on antimatroids. There are many equivalent axiomatizations of antimatroids, that may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. An algorthmic characterization of antimatroids, that considers them as set systems, was given in [4]. This characterization is based on the i...