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- W4298354290 abstract "For a presentation $mathcal{A}$ of a transversal matroid $M$, we study the set $T_{mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $mathcal{A}$; we order these extensions by the weak order. We show that $T_{mathcal{A}}$ is a distributive lattice, and that each finite distributive lattice is isomorphic to $T_{mathcal{A}}$ for some presentation $mathcal{A}$ of some transversal matroid $M$. We show that $T_{mathcal{A}}cap T_{mathcal{B}}$, for any two presentations $mathcal{A}$ and $mathcal{B}$ of $M$, is a sublattice of both $T_{mathcal{A}}$ and $T_{mathcal{B}}$. We prove sharp upper bounds on $|T_{mathcal{A}}|$ for presentations $mathcal{A}$ of rank less than $r(M)$ in the order on presentations; we also give a sharp upper bound on $|T_{mathcal{A}}cap T_{mathcal{B}}|$. The main tool we introduce to study $T_{mathcal{A}}$ is the lattice $L_{mathcal{A}}$ of closed sets of a certain closure operator on the lattice of subsets of ${1,2,ldots,r(M)}$." @default.
- W4298354290 created "2022-10-02" @default.
- W4298354290 creator A5040315652 @default.
- W4298354290 date "2015-08-04" @default.
- W4298354290 modified "2023-09-29" @default.
- W4298354290 title "Lattices related to extensions of presentations of transversal matroids" @default.
- W4298354290 doi "https://doi.org/10.48550/arxiv.1508.00870" @default.
- W4298354290 hasPublicationYear "2015" @default.
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