Varieties of combinatorial geometries

Abstract

A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class J \mathcal {J} of geometries is a sequence ( T n ) ({T_n}) of geometries in J \mathcal {J} with rank T n = n {T_n} = n , and satisfying the universal property: if G G is a geometry in J \mathcal {J} of rank n n , then G G is a subgeometry of T n {T_n} . A variety of geometries is a hereditary class with a sequence of universal models. We prove that, apart from two degenerate cases, the only varieties of combinatorial geometries are (1) the variety of free geometries, (2) the variety of geometries coordinatizable over a fixed finite field, and (3) the variety of voltage-graphic geometries with voltages in a fixed finite group.