Resolutions of convex geometries. (English) Zbl 1477.52001

Summary: Convex geometries [P. H. Edelman and R. E. Jamison, Geom. Dedicata 19, 247–270 (1985; Zbl 0577.52001)] are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions – compounds of hypergraphs, as in [M. Chein et al., Discrete Math. 37, 35–50 (1981; Zbl 0478.05071)], and compositions of set systems, as in [R. H. Möhring and F. J. Radermacher, Ann. Discrete Math. None, 257–356 (1984; Zbl 0567.90073)] –, resolutions of convex geometries always yield a convex geometry.
We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.


52A01 Axiomatic and generalized convexity
05B25 Combinatorial aspects of finite geometries
06A07 Combinatorics of partially ordered sets
51D20 Combinatorial geometries and geometric closure systems
Full Text: DOI arXiv


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