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Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems., Comment: 22 pages, 7 figures

Let H ? and K ? be finite composition series of length h in a group G . The intersections of their members form a lattice CSL ( H ? , K ? ) under set inclusion. Our main result determines the number N ( h ) of (isomorphism classes) of these lattices recursively. We also show that this number is asymptotically h ! / 2 . If the members of H ? and K ? are considered constants, then there are exactly h ! such lattices.Based on recent results of Czedli and Schmidt, first we reduce the problem to lattice theory, concluding that the duals of the lattices CSL ( H ? , K ? ) are exactly the so-called slim semimodular lattices, which can be described by permutations. Hence the results on h ! and h ! / 2 follow by simple combinatorial considerations. The combinatorial argument proving the main result is based on Czedli's earlier description of indecomposable slim semimodular lattices by matrices.