Matroids and antimatroids—a survey

Dietrich, Brenda

doi:10.1016/0012-365x(89)90180-5

Cited by 35 publications

(21 citation statements)

References 4 publications

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“…Note that (C1) and (C2) in Lemma 4.7 characterize the family of rooted circuits of a convex geometry among families of rooted subsets, that is, a given family C of rooted subsets of E satisfies (C1) and (C2) if and only if C is the family of rooted circuits of some convex geometry on E. This characterization is due to Dietrich [4,5].…”

Section: Lemma 47mentioning

confidence: 96%

“…In the literature [2,5,7,15] the reader can find more theory of convex geometries (or antimatroids, equivalently). For the sake of completeness of the paper, we will include the proofs of most of the lemmas so that we can get some intuitions about these concepts with which the reader may be unfamiliar.…”

Section: More Concepts From Convex Geometriesmentioning

confidence: 99%

“…A convex geometry was introduced by Edelman and Jamison [7] as an abstraction of convexity, and it can be seen as a "dual" (or a "polar" or a "complement") of an antimatroid [5]. (Therefore, we sometimes use the word "antimatroid" instead of "convex geometry" to express the same object.)…”

Section: Introductionmentioning

confidence: 99%

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The affine representation theorem for abstract convex geometries

Kashiwabara

Nakamura

Okamoto

2005

Computational Geometry

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A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of "convexity" shared by some objects including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and show that every convex geometry can be represented as a generalized convex shelling. This is "the representation theorem for convex geometries" analogous to "the representation theorem for oriented matroids" by Folkman and Lawrence. An important feature is that our representation theorem is affine-geometric while that for oriented matroids is topological. Thus our representation theorem indicates the intrinsic simplicity of convex geometries, and opens a new research direction in the theory of convex geometries.

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Section: Lemma 47mentioning

confidence: 96%

Section: More Concepts From Convex Geometriesmentioning

confidence: 99%

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The affine representation theorem for abstract convex geometries

Kashiwabara

Nakamura

Okamoto

2005

Computational Geometry

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“…Finally, numerous examples of anti-matroids, whose closure will yield a closure space, can be found in the survey of anti-matroids [7] or the text on greedoids [15] which generalize an important class of computer algorithms.…”

Section: Closure Spacesmentioning

confidence: 99%

“…is the closure operator of an anti-matroid [7] [15]. In [21] it is shown that Closure spaces are fairly common in computer science and its applications, although they frequently have other names.…”

Section: Closure Spacesmentioning

confidence: 99%

Partition coefficients of acyclic graphs

Pfaltz

1995

Graph-Theoretic Concepts in Computer Science

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Abstract. We develop the concept of a "closure space" which appears with different names in many aspects of graph theory. We show that acyclic graphs can be almost characterized by the partition coefficients of their associated closure spaces. The resulting nearly total ordering of all acyclic graphs (or partial orders) provides an effective isomorphism filter and the basis for efficient retrieval in secondary storage. Closure spaces and their partition coefficients provide the theoretical basis for a new computer system being developed to investigate the properties of arbitrary acyclic graphs and partial orders. Binary PartitionsIn this paper we combine two mathematical threads and apply them in a graphtheoretic context. The first thread of binary partitions was studied by Euler as early as 1750. The second thread involving closure spaces is of more recent origin. A binary partition of a positive integer N is its expression as a sum of powers of 2. Mahler [16], and Churchhouse [3] [4] have studied binary partitions from a number theoretic point of view. Because our intention is to connect these partitions with closure spaces, we will confine our attention to the special case where N is also a power of 2.By a binary partition of 2 n we mean a sequence of non-negative integers <.-.,ak.-.>,0 0 must be even, the coefficient a0 must be even. Third, if < ...,ak,ak-1,... > is a partition, then < ...,ak -1, ak_l + 2,-.. > must be as well. And fourth, if < an,...,ak,...,ao > is a partition of 2 ~ then < as, -9 ", ak, 9 9 a0, 0 > is a partition of 2 n+l .With these observations, it is not difficult to write a process which generates all partitions in lexicographic order. Doing so, and displaying each partition,

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A Tutte Polynomial for Partially Ordered Sets

Gordon

1993

Journal of Combinatorial Theory, Series B

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Matroids and antimatroids—a survey

Cited by 35 publications

References 4 publications

The affine representation theorem for abstract convex geometries

The affine representation theorem for abstract convex geometries

Partition coefficients of acyclic graphs

A Tutte Polynomial for Partially Ordered Sets

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