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 t...

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Bibliographic Details
Published in:European journal of combinatorics 2014-04, Vol.37, p.68-78
Main Author: Adaricheva, Kira
Format: Article
Language:English
Subjects:
Axioms
Closures
Combinatorial analysis
Lattices
Mathematical analysis
Partitions
Strengthening
Vector spaces
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Summary: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 finite sub-geometries satisfy the n-Carousel Rule, which is the strengthening of the n-Carathéodory property. We also find another property, that is similar to the simplex partition property and independent of 2-Carousel Rule, which holds in sub-geometries of 2-dimensional geometries of relatively convex sets.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2013.07.012