Varieties and universal models in the theory of combinatorial geometries

Abstract

One of the most attractive but little studied ideas in the theory of combinatorial geometries (or matroids) [2], [4] is the notion of a hereditary class of geometries. A hereditary class of (finite) geometries is a collection of geometries which is closed under taking minors and direct sums. Thus, hereditary classes are direct analogues of varieties in universal algebra. Although varieties are highly structured collections of objects-free objects exist, for example-hereditary classes are relatively unstructured. In order to obtain reasonable results, it is necessary to impose some regularity conditions. One possibility is to postulate the existence of free (or rather, cofree) objects. A sequence of universal models for a hereditary class T of geometries is a sequence (T n ) of geometries in T with rank (T n ) = n satisfying the universal property: if G is a geometry in T of rank w, then G is a subgeometry of T n . A variety of geometries is a hereditary class with a sequence of universal models. Rather surprisingly, it is possible to classify varieties of geometries.Before stating our result, we need to describe two simple sequences of geometries. Let Afj be the rank one geometry and M 2 be the line with q + 1 points.The subgeometries of these geometries form a variety called the variety of matchstick geometries of order q. Now, let B n be the Boolean algebra on the set {1,...,«}. On each of the lines Ï2,23,..., (n -\)n, add q -1 points in general position and call the resulting geometry O n (q). The subgeometries of these geometries also form a variety called the variety of origami geometries of order q.THEOREM . Let T be a variety of geometries. Then, T is one of the following collections:1. the variety of free geometries; 2. the variety of matchstick geometries of order q\ 3. the variety o f origami geometries of order q\ 4. the variety of all geometries coordinatizable over a fixed finite field GF(q); 5. the variety of voltage-graphic geometries with voltages in a fixed finite group A.