{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T06:58:43Z","timestamp":1760597923821,"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>We call a quadruple $\\mathcal{W}:=\\langle F,U,\\Omega,\\Lambda \\rangle$, where $U$ and $\\Omega$ are two given non-empty finite sets, $\\Lambda$ is a non-empty set and $F$ is a map having domain $U\\times \\Omega$ and codomain $\\Lambda$, a pairing\u00a0on $\\Omega$. With this structure we associate a set operator $M_{\\mathcal{W}}$ by means of which it is possible to define a preorder $\\ge_{\\mathcal{W}}$ on the power set $\\mathcal{P}(\\Omega)$ preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice $\\mathbb{L}$ there exist a finite set $\\Omega_{\\mathbb{L}}$ and a pairing $\\mathcal{W}$ on $\\Omega_\\mathbb{L}$ such that the quotient of the preordered set $(\\mathcal{P}(\\Omega_\\mathbb{L}), \\ge_\\mathcal{W})$ with respect to its symmetrization is a lattice that is order-isomorphic to $\\mathbb{L}$. In the second result, we prove that when the lattice $\\mathbb{L}$ is endowed with an order-reversing involutory map $\\psi: L \\to L$ such that $\\psi(\\hat 0_{\\mathbb{L}})=\\hat 1_{\\mathbb{L}}$, $\\psi(\\hat 1_{\\mathbb{L}})=\\hat 0_{\\mathbb{L}}$, $\\psi(\\alpha) \\wedge \\alpha=\\hat 0_{\\mathbb{L}}$ and $\\psi(\\alpha) \\vee \\alpha=\\hat 1_{\\mathbb{L}}$, there exist a finite set $\\Omega_{\\mathbb{L},\\psi}$ and a pairing on it inducing a specific poset which is order-isomorphic to $\\mathbb{L}$.<\/jats:p>","DOI":"10.37236\/8786","type":"journal-article","created":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T09:18:24Z","timestamp":1579857504000},"source":"Crossref","is-referenced-by-count":8,"title":["Lattice Representations with Set Partitions Induced by Pairings"],"prefix":"10.37236","volume":"27","author":[{"given":"Giampiero","family":"Chiaselotti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tommaso","family":"Gentile","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Federico","family":"Infusino","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,1,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p19\/8004","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p19\/8004","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T09:18:24Z","timestamp":1579857504000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8786","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,1,24]]},"article-number":"P1.19"}}