Discrete Mathematics 308 (2008) 4734 – 4744 www.elsevier.com/locate/disc Rank functions of strict cg-matroids Yoshio Sano Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Received 26 July 2006; received in revised form 23 August 2007; accepted 24 August 2007 Available online 3 December 2007 Abstract A matroid-like structure defined on a convex geometry, called a cg-matroid, is defined by Fujishige et al. [Matroids on convex geometries (cg-matroids), Discrete Math. 307 (2007) 1936–1950]. A cg-matroid whose rank function is naturally defined is called a strict cg-matroid. In this paper, we give characterizations of strict cg-matroids by their rank functions. © 2007 Elsevier B.V. All rights reserved. Keywords: Matroid; Convex geometry; Rank function 1. Introduction A matroid is one of the most important structures in combinatorial optimization. Many researchers have studied and extended the matroid theory. Dunstan et al. [3] introduced the concept of a supermatroid defined on a poset in 1972 as a generalization of the concept of an ordinary matroid ([14]; also see [13,10]). Faigle defined submodular supermatroids [7] in 1978, and considered a geometric structure on a poset [6] in 1980. Tardos [12] showed a matroid-type intersection theorem for distributive supermatroids in 1990. A distributive supermatroid is also called a poset matroid. Peled and Srinivasan [11] considered a matroid-type independent matching problem for poset matroids in 1993. Moreover, in 1993 and 1998 Barnabei et al. [2,1] studied poset matroids in terms of the poset structure of the ground set. In [9], Fujishige et al. generalized poset matroids by considering convex geometries, instead of posets, as underlying combinatorial structures on which they define matroid-like structures, called cg-matroids. For a cg-matroid they defined independent sets, bases, and other related concepts, and examined their combinatorial structural properties. They have shown characterizations of the families of bases, independent sets, and spanning sets of cg-matroids. It is shown that cg-matroids are not special cases of supermatroids. They also considered a special class of cg-matroids, called strict cg-matroids, for which rank functions are naturally defined, and they show the equivalence of the concept of a strict cg-matroid and that of a supermatroid defined on the lattice of closed sets of a convex geometry. (See Fig. 1.) The rank functions of strict cg-matroids were defined. And they have shown some properties which the rank functions satisfy. But it was unknown to characterize strict cg-matroids in terms of rank functions. In this paper, we give characterizations of the rank functions of strict cg-matroids. Our main results are as follows. Let Z + be the set of nonnegative integers. E-mail address: sano@kurims.kyoto-u.ac.jp. 0012-365X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2007.08.095