{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:20:28Z","timestamp":1758824428979,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this article, we investigate the set of $\\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\\gamma\\in W$, under Bruhat order, and we denote this poset by $\\mathcal{B}_{\\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and M\u00e9sz\u00e1ros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\\mathcal{B}_{\\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\\gamma\\in W$ the lattice $\\mathcal{B}_{\\gamma}$ is in fact distributive. It turns out that this is the case for the \"coincidental\" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\\gamma$ of said groups for which $\\mathcal{B}_{\\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.<\/jats:p>","DOI":"10.37236\/4942","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:49:14Z","timestamp":1578700154000},"source":"Crossref","is-referenced-by-count":2,"title":["SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements"],"prefix":"10.37236","volume":"22","author":[{"given":"Henri","family":"M\u00fchle","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,6,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p40\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:16:03Z","timestamp":1579256163000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i2p40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/4942","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,6,3]]},"article-number":"P2.40"}}