Modular constructions for combinatorial geometries

Abstract

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial χ ( x ) \chi (x) of a modular flat x x divides the characteristic polynomial χ ( G ) \chi (G) of a geometry G G . In this paper we identify the quotient: THEOREM. If x x is a modular flat of G , χ ( G ) / χ ( x ) = χ ( T x ¯ ( G ) ) / ( λ − 1 ) G,\chi (G)/\chi (x) = \chi (\overline {{T_x}} (G))/(\lambda - 1) , where T x ¯ ( G ) \overline {{T_x}} (G) is the complete Brown truncation of G G by x x . (The lattice of T x ¯ ( G ) \overline {{T_x}} (G) consists of all flats containing x x and all flats disjoint from x x , with the induced order from G G .) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries G G and H H , if x x is a modular flat of G G as well as a subgeometry of H H , then there exists a geometry P = P x ( G , H ) P = {P_x}(G,H) which is a pushout in the category of injective strong maps and such that χ ( P ) = χ ( G ) χ ( H ) / χ ( x ) \chi (P) = \chi (G)\chi (H)/\chi (x) . The closed set structure, rank function, independent sets, and lattice properties of P P are characterized. After proving a modular extension theorem we give applications of our results to Crapo’s single element extension theorem, Crapo’s join operation, chain groups, unimodular geometries, transversal geometries, and graphs.