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Combinatorial flag arrangements

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arxiv 2404.01971 v1 pith:RDSR5VRT submitted 2024-04-02 math.CO math.AG

classification math.COmath.AG
keywords combinatorialarrangementsmatricubesmatroidsaxiomatizationarraysaxiomaticcircuits
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We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of arrangements of initial flags in a vector space. We give cryptomorphic axiomatic systems in terms of rank function, flats, circuits, and independent sets, and formulate a duality concept. We also provide precise links between matricubes, permutation arrays and matroids, and raise several open questions.

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  1. Relating Different Definitions of Linear Series on Tropical Curves

    math.AG 2025-06 conditional novelty 7.0 of 10

    A tropical linear series is combinatorial limit if and only if it is structured, and every strongly recursive tropical linear series is combinatorial limit; the reverse inclusion fails from rank three onward.

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