Séminaire Lotharingien de Combinatoire, B19h (1988).
[Formerly: Publ. I.R.M.A. Strasbourg, 1988, 361/S-19, p. 138-139.]

Walter Wenzel

Matroidizing Set Systems

Abstract. A matroid M(B) is associated in a canonical way to every antichain B of a finite nonempty set E. For this purpose, a sequence of alternate derivations of closure operators and antichains on E is introduced: the initial antichain is B; the closure (derived from an antichain B) of a set X consists of all those elements e of E which can be replaced by an element of X in all the sets of B containing e so that another set of B is produced; the antichain derived from a closure operator consists of all the minimal generating sets of the latter.

It is proved that the deriving process stops after a finite number of steps (i.e., there necessarily exists a fixed point). The final antichain and the closure operator are the family of bases and the closure operator of the matroid M(B).

The paper has been finally published as a joint paper with Andreas Dress under the title "Matroidizing set systems: a new approach to matroid theory" in Appl. Math. Lett. 3 (1990), 29-32.