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