{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:28Z","timestamp":1753893808012,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a presentation $\\mathcal{A}$ of a transversal matroid $M$, we study the ordered set $T_{\\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\\mathcal{A}$; extensions are ordered by the weak order.\u00a0 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\u00a0$T_{\\mathcal{A}}$ 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\u00a0$T_{\\mathcal{A}}$ is the lattice\u00a0$L_{\\mathcal{A}}$ of closed sets of a certain closure operator on the lattice of subsets of $\\{1,2,\\ldots,r(M)\\}$.<\/jats:p>","DOI":"10.37236\/5466","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:09:47Z","timestamp":1578683387000},"source":"Crossref","is-referenced-by-count":0,"title":["Lattices Related to Extensions of Presentations of Transversal Matroids"],"prefix":"10.37236","volume":"24","author":[{"given":"Joseph E.","family":"Bonin","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,3,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:03:40Z","timestamp":1579237420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/5466","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,3,17]]},"article-number":"P1.49"}}