Weak cuts of combinatorial geometries
Hien Nguyen · Transactions of the American Mathematical Society · 1979
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 M {\mathcal {M}} of independent sets of G , is associated a unique weak cut C {\mathcal {C}} containing M {\mathcal {M}} . In practice, the flats of the weak image defined by C {\mathcal {C}} 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 F -erection of G is an erection of G which is a weak image of F . The main results are that the set of all F -erections of G is a lattice with the weak map order, and that the free F -erection can be constructed explicitly. Finally, a problem involving higher order erection is solved.