Weak Cuts of Combinatorial Geometries

A weak cut of a Combinatorial Geometry G is a generalization of a modular cut, corresponding to the family of the new dependent sets in a weak map image of G. The use of weak cuts allows the construction of all weak images of G, an important result being that, to any family "311 of independent sets of G, is associated a unique weak cut S containing "311. In practice, the flats of the weak image defined by 6 can be constructed directly. The weak cuts corresponding to known weak maps, such as truncation, projection, elementary quotient, are determined. The notion of weak cut is particularly useful in the study of erections. Given a geometry F and a weak image G, an /"-erection of G is an erection of G which is a weak image of F. The main results are that the set of all /•"-erections of G is a lattice with the weak map order, and that the free /•"-erection can be constructed explicitly. Finally, a problem involving higher order erection is solved. Introduction. In the theory of Combinatorial Geometries, whereas strong maps have been widely studied, much fewer research papers have focused on the more general concept of weak maps. After the pioneering work of Higgs [5], [6] on maps of geometries, Lucas' thesis [9] was probably the first study to concentrate specially on problems concerning weak maps. In addition to their intrinsic interest, the study of weak maps is highly relevant to several important questions and open conjectures in the general theory of Combinatorial Geometries, as is well shown in [9]. It can also be noted that examples of weak maps abound in algebraic geometry: for example, consider the specialization of a point in general position in 3-space onto a plane or a straight line. In this paper, we consider the problem of constructing weak map images and preimages of a given geometry, with the view of developing a technique useful for analyzing problems related to weak maps in general. A fundamental procedure in the study of strong maps is to use the one-to-one correspondence between strong maps and modular cuts. The main approach in this paper is to introduce a more general notion of cuts, called weak cuts, for the case of weak maps. Received by the editors July 25, 1977 and, in revised form, April 14, 1978. AMS (A/OS) subject classifications (1970). Primary 05B25, 05B35; Secondary 05A20, 06A20.