Pattern Recognition on Oriented Matroids

Matveev, Andrey O.

doi:10.1515/9783110531145

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…Blocking sets and the blockers of set families (families are often regarded as the hyperedge families of hypergraphs) are discussed, e.g., in the monographs [11,22,29,31,32,43,46,48,50,51,53,54,56,67,70,72,73,78,79,81] and in the works [3,4,5,6,7,9,10,12,13,16,17,20,21,23,24,25,26,27,28,30,33,34,38,39,40,41,42,44,45,47,52,…”

Section: Blockingmentioning

confidence: 99%

Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V

Matveev

2021

Preprint

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We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in R 2 t , with respect to a distinguished symmetric 2 • 2 t -cycle in its hypercube graph of topes H(2 t , 2). We seek interpretations of such decompositions in the context of subset families on the ground set Et := {1, . . . , t} and of the families of their blocking sets, in the context of clutters on Et and of their blockers. Contents 1. Introduction Decomposing 2. Topes, their relabeled opposites, and decompositions Blocking 3. Increasing families of blocking sets, and blockers: Set covering problems 4. Families of subsets of the ground set E t : Characteristic vectors and characteristic topes 5. Increasing families of blocking sets, and blockers: Characteristic vectors and characteristic topes 5.1. A clutter {{a}} 5.1.1. The principal increasing family of blocking sets B({{a}}) = {{a}} 5.1.2. The blocker B({{a}}) = {{a}} 5.1.3. More on the principal increasing family B({{a}}) = {{a}} 5.2. A clutter {A} 5.2.1. The increasing family of blocking sets B({A}) = {{a} : a ∈ A} 5.2.2. The blocker B({A}) = {{a} : a ∈ A} 5.2.3. More on the increasing families {A} and B({A}) 5.3. A clutter A := {A 1 , . . . , A α } 5.3.1. The increasing family of blocking sets B(A) 5.3.2. The blocker B(A) 5.3.3. More on the increasing families A and B(A)

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Section: Blockingmentioning

confidence: 99%

Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V

Matveev

2021

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“…A subset B ⊆ E t is a blocking set of a nontrivial clutter A ⊂ 2 [t] if it holds |B ∩ A| > 0, for each member A ∈ A. The blocker B(A) of the clutter A is defined to be the family of all inclusion-minimal blocking sets of A; see, e.g., the monographs [4,7,9,10,11,12,13,15,16,19,20,21,22,24,26,31,32,33,35,36,38]. Additional references can be found in [27,Pt 2].…”

Section: Introductionmentioning

confidence: 99%

The Increasing Families of Sets Generated by Self-dual Clutters

Matveev¹

2021

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“…Recall that the vertex set V(R) of the cycle R is the set of topes of a rank 2 oriented matroid denoted A R , see e.g. [3,Rem. 1.7], and thus by [1,Example 7.1.7] we can consider the representation of A R by a certain central line arrangement…”

Section: Introductionmentioning

confidence: 99%

“…If the tope graph T (L(A)) is the hypercube graph H(t, 2) on 2 t vertices, then for any odd integer j, 1 ≤ j ≤ t, by [3,Th. 13.6] there are precisely 2 t j vertices T of H(t, 2) such that |Q(T, R)| = j.…”

Section: Introductionmentioning

confidence: 99%