Functions of the category of combinatorial geometries and strong maps

As in the case with most mathematical structures, an important question in the theory of combinatorial geometries is to develop constructions for obtaining new geometries from old ones. Several geometric constructions are well-known, e.g. deletion, contraction, truncation, direct sums, etc. . . . and are generally extensions to combinatorial geometries of existing operations on projective geometries, graphs or lattices. In this paper we introduce a new class o’: constructions based on the ideas of expansion and geomctrization. More precisely, we will study several classes of funccors of the category Y of combinatorial pregeometries and strong maps into itself. The motivation of this study was the following conjecture of Rota: Given a pregeometry C(r, s) and an integer k 5 I, the geometrization of the function kr is a functor of Y into itself. After a brief review of the basic concepts of the theory of combinatorial geometries in Section 2, the conjecture is proved in Section 3 as a consequence of d general study concerning a broader class of functors. In Section 3. a generalization is considered which yields a method for constructing new classes of functors.