Description of closure operators in convex geometries of segments on the line. (English) Zbl 1423.05003

Summary: Convex geometry is a closure space \((G,\phi)\) with the anti-exchange property. A classical result of P. H. Edelman and R. E. Jamison [Geom. Dedicata 19, 247–270 (1985; Zbl 0577.52001)] claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary and sufficient conditions on a closure operator \(\phi\) of convex geometry \((G,\phi)\) so that its convex dimension equals 2, equivalently, they are represented by segments on a line. These conditions, for a given convex geometry \((G,\phi)\), can be checked in polynomial time in two parameters: the size of the base set \(|G|\) and the size of the implicational basis of \((G,\phi)\).


05A05 Permutations, words, matrices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B99 Lattices
52B55 Computational aspects related to convexity


Zbl 0577.52001
Full Text: DOI arXiv


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