#####
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.