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Perfect matroids

Authors
Journal
Advances in Mathematics
0001-8708
Publisher
Elsevier
Publication Date
Volume
91
Issue
2
Identifiers
DOI: 10.1016/0001-8708(92)90015-d
Disciplines
  • Mathematics

Abstract

Abstract Matroids with an arbitrary domain of coefficients have been introduced in [A. W. M. Dress, Adv. Math. 59 (1986), 97–123] and, since, studied in [A. W. M. Dress and W. Wenzel, Adv. Math. 77 (1989), 1–36; Adv. Math. 86 (1991), 68–110; Bayreuth. Math. Schr. 26 (1988), 37–98; Geom. Dedicata 34 (1990), 161–197; Appl. Math. Lett. 3, No. 2 (1990), 33–35; Adv. Math., in press; M. Wagowski, European J. Combin. 10 (1989), 393–398; W. Wenzel, Adv. Math. 77 (1989), 37–75; J. Combin. Theory Ser. A 57 (1991), 15–45]. In the present paper we study such matroids whose coefficients belong to a particular, but rather natural, class of such domains, the so-called perfect fuzzy rings. These include matroids representable over a ring as well as ordinary, oriented, and valuated matroids. A number of well-known and important results which are known to hold for such matroids (e.g., Tutte's representability theorem and the corresponding results for oriented and valuated matroids), but do not hold for arbitrary matroids with coefficients, as well as some additional results concerning, e.g., (fuzzy) determinant identities, which were not known even in the case of oriented matroids, are shown to hold more generally for matroids with coefficients in perfect domains.

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