A Combinatorial Geometric Structure on the Space of Orders of a Field: I

We introduce a structure of combinatorial geometry on the setχ(K)of orders of an orderable fieldK.It verifies finiteness conditions allowing us to develop a smooth combinatorial theory of the spacesχ(K)regardless of whether they are finite or infinite. It is shown that the geometriesχ(K)are binary and bipartite. We introduce the combinatorial notion of afanand use it to study binary, bipartite combinatorial geometries in general. We characterize the fieldsKfor which the geometryχ(K)is unimodular; this extends a classical result of Bröcker in quadratic form theory.