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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.

MSC:
05B99 Designs and configurations
05B35 Combinatorial aspects of matroids and geometric lattices
Keywords:
duality; matroids
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[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
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