  Convex geometries (Edelman and Jamison, 1985) 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 Chein, Habib and
Maurer (1981), and compositions of set systems, as in Mohring and Radermacher
(1984) -- , 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.
