zbMATH — the first resource for mathematics

An abstract duality. (English) Zbl 0686.05014
The paper deals with a concept of abstract duality defined on a family of structures (matroids, Sperner families, oriented matroids, for instance) as a correspondence among members of the family that is an involution, preserve the ground set and interchanges contractions and deletions. The uniqueness of such a correspondence for several structures is discussed. For a more detailed study of the proofs, the authors refer to the Ph. D. thesis of the second named and a joint technical report published by the Cornell University School of OR/IE (No.726) and IBM (No.12527) in 1986 and 1987, respectively.

05B99 Designs and configurations
05B35 Combinatorial aspects of matroids and geometric lattices
duality; matroids
Full Text: DOI
[1] Bland, R.G.; Dietrich, B.L., A unified interpretation of several combinatorial dualities, Cornell university school of OR/IE technical report no. 726, IBM research report 12527, (1987), also appears as
[2] Bland, R.G.; Jensen, D.L., Weakly oriented matroids, ()
[3] Bland, R.G.; Las Vergnas, M., Orientability of matroids, J. combin. theory ser. B, 24, 94-134, (1978) · Zbl 0374.05016
[4] Dietrich, B.L., A unifying interpretation of several combinatorial dualities, () · Zbl 1190.90156
[5] Dietrich, B.L., A circuit set characterization of antimatroids, J. combin. theory ser. B, 43, 314-321, (1987) · Zbl 0659.05036
[6] Edelman, P.H., Meet-distributive lattices and anti-exchange closure, Algebra univ., 10, 463-470, (1980) · Zbl 0442.06004
[7] Edelman, P.H.; Jamison, R., The theory of convex geometries, Geom. dedicata, 19, 247-274, (1985) · Zbl 0577.52001
[8] Edmonds, J.; Fulkerson, D.R., Bottleneck extrema, J. combin. theory, 8, 299-306, (1970) · Zbl 0218.05006
[9] Fulkerson, D.R., Blocking and anti-blocking pairs of polyhedra, Math. programming, 1, 168-194, (1971) · Zbl 0254.90054
[10] Jamison-Waldner, R.E., A perspective on abstract convexity: classifying alignments by varieties, (), 113-150 · Zbl 0482.52001
[11] Jensen, D., Coloring and duality: combinatorial augmentation methods, ()
[12] Korte, B.; Lovász, L., Shelling structures, convexity, and a happy end, (), 219-232 · Zbl 0553.05030
[13] Kung, J.P.S., A characterization of orthogonal duality in matroid theory, Geom. dedicata, 15, 69-72, (1983) · Zbl 0531.05027
[14] Welsh, D.J.A., Matroid theory, (1976), Academic Press New York · Zbl 0343.05002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.