Resolutions of convex geometries (Q2665967)

Summary: Convex geometries [\textit{P. H. Edelman} and \textit{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 [\textit{M. Chein} et al., Discrete Math. 37, 35--50 (1981; Zbl 0478.05071)], and compositions of set systems, as in [\textit{R. H. Möhring} and \textit{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.