Matroidizing Set Systems
Séminaire lotharingien de combinatoire, Tome 19 (1988)
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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.
@article{SLC_1988_19_a7,
author = {Walter Wenzel},
title = {Matroidizing {Set} {Systems}},
journal = {S\'eminaire lotharingien de combinatoire},
publisher = {mathdoc},
volume = {19},
year = {1988},
url = {http://geodesic.mathdoc.fr/item/SLC_1988_19_a7/}
}
Walter Wenzel. Matroidizing Set Systems. Séminaire lotharingien de combinatoire, Tome 19 (1988). http://geodesic.mathdoc.fr/item/SLC_1988_19_a7/