Authors: Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: A Glimpse into Continuous Combinatorics of Posets, Polytopes, and Matroids
Journal: Journal of Mathematical Sciences (United States)
Volume: 248
Issue: 6
First page: 762
Last page: 775
Issue Date: 1-Aug-2020
Rank: M51
ISSN: 1072-3374
DOI: 10.1007/s10958-020-04910-1
Abstract: 
We advocate a systematic study of continuous analogs of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples reviewed are an Euler formula for a class of “continuous convex polytopes” (conjectured by Kalai and Wigderson), a duality result for a class of “continuous matroids,” a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of G.-C. Rota), an exposition of the “homotopy complementation formula” for topological posets and its relation to the results of S. Kallel and R. Karoui about “weighted barycenter spaces,” and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria’s inequality) based on interpreting diagrams of spaces as continuous posets.
Publisher: Springer Link

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