Elementary Maps Research Articles

Elementary strong maps and transversal geometries

Let H→G be a strong map between two combinatorial geometries on the same set X. The rank function, flats, and independent sets of G are characterized in terms of a factorization of E→G into elementary strong maps. When H is the free geometry on X, these results lead to a representation of G as the “basis intersection” of a family of transversal geometries (in the sense that B is a basis for G if and only if B is a basis for each of the transversal geometries in the family), and dually, as the basis intersection of a family of principal geometries.

Elementary strong maps and transversal geometries

Let H → G be a strong map between two combinatorial geometries on the same set X . The rank function, flats, and independent sets of G are characterized in terms of a factorization of E → G into elementary strong maps. When H is the free geometry on X , these results lead to a representation of G as the “basis intersection” of a family of transversal geometries (in the sense that B is a basis for G if and only if B is a basis for each of the transversal geometries in the family), and dually, as the basis intersection of a family of principal geometries.