{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T22:17:32Z","timestamp":1649197052259},"reference-count":18,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:p> We prove that for every distributive \u3008\u2228,0\u3009-semilattice S, there are a meet-semilattice P with zero and a map \u03bc: P \u00d7 P \u2192 S such that \u03bc(x,z) \u2264 \u03bc(x,y) \u2228 \u03bc(y,z) and x \u2264 y implies that \u03bc(x,y) = 0, for all x, y, z \u2208 P, together with the following conditions:. <\/jats:p><jats:p> (P1) \u03bc(v,u) = 0 implies that u = v, for all u \u2264 v in P. <\/jats:p><jats:p> (P2) For all u \u2264 v in P and all a, b \u2208 S, if \u03bc(v,u) \u2264 a \u2228 b, then there are a positive integer n and a decomposition u = x<jats:sub>0<\/jats:sub> \u2264 x<jats:sub>1<\/jats:sub> \u2264 \u22ef \u2264 x<jats:sub>n<\/jats:sub> = v such that either \u03bc(x<jats:sub>i + 1<\/jats:sub>, x<jats:sub>i<\/jats:sub>) \u2264 a or \u03bc(x<jats:sub>i+1<\/jats:sub>, x<jats:sub>i<\/jats:sub>) \u2264 b, for each i &lt; n. <\/jats:p><jats:p> (P3) The subset {\u03bc(x,0) | x \u2208 P} generates the semilattice S. <\/jats:p><jats:p> Furthermore, every finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive \u3008\u2228,0,1\u3009-semilattices. <\/jats:p>","DOI":"10.1142\/s0218196708004469","type":"journal-article","created":{"date-parts":[[2008,3,20]],"date-time":"2008-03-20T09:29:16Z","timestamp":1206005356000},"page":"321-356","source":"Crossref","is-referenced-by-count":1,"title":["POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES"],"prefix":"10.1142","volume":"18","author":[{"given":"FRIEDRICH","family":"WEHRUNG","sequence":"first","affiliation":[{"name":"CNR, UMR 6139, D\u00e9partement de Math\u00e9matiques, BP 5186, Universit\u00e9 de Caen, Campus 2, 14032 Caen Cedex, France"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01455948"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.3792\/pia\/1195573786"},{"key":"rf4","unstructured":"G.\u00a0Gr\u00e4tzerB. A.\u00a0Davey, General Lattice Theory, 2nd edn. (Birkh\u00e4user Verlag, Basel, 1998)\u00a0p. xx+663."},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-005-1889-1"},{"key":"rf6","first-page":"34","volume":"24","author":"Gr\u00e4tzer G.","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10377"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1982.103.475"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01236809"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-006-1962-4"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2006.11.005"},{"key":"rf12","first-page":"153","volume":"43","author":"Schmidt E. T.","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196701000516"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196706003049"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-99-04558-X"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-8693(03)00015-2"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2005.02.009"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-005-1951-z"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2007.05.016"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196708004469","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T22:28:51Z","timestamp":1565130531000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196708004469"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":18,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2008,3]]}},"alternative-id":["10.1142\/S0218196708004469"],"URL":"https:\/\/doi.org\/10.1142\/s0218196708004469","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3]]}}}