{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:14:32Z","timestamp":1758824072999},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2009,9,1]],"date-time":"2009-09-01T00:00:00Z","timestamp":1251763200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:p>We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colourings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice, this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on \u0394-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular,<jats:italic>c<\/jats:italic>-orientations (Propp), \u03b1-orientations of planar graphs (Felsner, resp. de Mendez) and planar flows (Khuller, Naor and Klein). The characterization also applies to other instances,<jats:italic>e.g.<\/jats:italic>, to chip-firing games.<\/jats:p>","DOI":"10.1017\/s0963548309010001","type":"journal-article","created":{"date-parts":[[2009,6,1]],"date-time":"2009-06-01T09:21:11Z","timestamp":1243848071000},"page":"707-724","source":"Crossref","is-referenced-by-count":13,"title":["ULD-Lattices and \u0394-Bonds"],"prefix":"10.1017","volume":"18","author":[{"given":"STEFAN","family":"FELSNER","sequence":"first","affiliation":[]},{"given":"KOLJA B.","family":"KNAUER","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2009,9,1]]},"reference":[{"key":"S0963548309010001_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(13)80111-4"},{"key":"S0963548309010001_ref19","first-page":"219","volume-title":"Graph Theory and Combinatorics","author":"Korte","year":"1984"},{"key":"S0963548309010001_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2008.08.020"},{"key":"S0963548309010001_ref16","first-page":"534","article-title":"Copoints in antimatro\u00efds","volume":"29","author":"Jamison","year":"1980","journal-title":"Congr. Numer."},{"key":"S0963548309010001_ref18","unstructured":"[18] Knauer K. B. (2007) Partial orders on orientations via cycle flips. http:\/\/www.math.tu-berlin.de\/~knauer\/diplom.pdf."},{"key":"S0963548309010001_ref15","first-page":"15","article-title":"A development of axiomatic convexity","volume":"48","author":"Jamison","year":"1970","journal-title":"Clemson University Math."},{"key":"S0963548309010001_ref8","unstructured":"[8] Boulaye G. (1967) Sous-arbres et homomorphismes \u00e0 classes connexes dans un arbre. In Theory of Graphs International Symposium, pp. 47\u201350."},{"key":"S0963548309010001_ref14","unstructured":"[14] Felsner S. and Knauer K. B. (2008) Distributive lattices, polyhedra, and generalized flow. arXiv:0811.1541v1[math.CO]."},{"key":"S0963548309010001_ref4","doi-asserted-by":"publisher","DOI":"10.2307\/1999881"},{"key":"S0963548309010001_ref3","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF00333128","article-title":"The convexity lattice of a poset","volume":"2","author":"Birkhoff","year":"1985","journal-title":"Order"},{"key":"S0963548309010001_ref23","unstructured":"[23] Propp J. (1993) Lattice structure for orientations of graphs. arXiv:math\/0209005v1[math.CO]."},{"key":"S0963548309010001_ref5","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022467132614"},{"key":"S0963548309010001_ref10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1960-0118690-9"},{"key":"S0963548309010001_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02482912"},{"key":"S0963548309010001_ref26","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511665578"},{"key":"S0963548309010001_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662041.009"},{"key":"S0963548309010001_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(71)90074-8"},{"key":"S0963548309010001_ref13","doi-asserted-by":"crossref","DOI":"10.37236\/1768","article-title":"Lattice structures from planar graphs","volume":"11","author":"Felsner","year":"2004","journal-title":"Electron. J. Combin."},{"key":"S0963548309010001_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF00149365"},{"key":"S0963548309010001_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(02)00581-4"},{"key":"S0963548309010001_ref24","doi-asserted-by":"crossref","DOI":"10.37236\/1330","article-title":"Generating random elements of finite distributive lattices","volume":"4","author":"Propp","year":"1997","journal-title":"Electron. J. 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