{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T07:49:54Z","timestamp":1771573794393,"version":"3.50.1"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2008,3,1]],"date-time":"2008-03-01T00:00:00Z","timestamp":1204329600000},"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":[[2008,3]]},"abstract":"<jats:p>Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group <jats:italic>S<jats:sub>n<\/jats:sub><\/jats:italic>. In this paper we resolve two problems on semi-graphoids posed in Studen\u00fd's book (2005), and we answer a related question of Postnikov, Reiner and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semi-graphoids.<\/jats:p>","DOI":"10.1017\/s0963548307008838","type":"journal-article","created":{"date-parts":[[2007,12,7]],"date-time":"2007-12-07T04:42:19Z","timestamp":1197002539000},"page":"239-257","source":"Crossref","is-referenced-by-count":11,"title":["Three Counter-Examples on Semi-Graphoids"],"prefix":"10.1017","volume":"17","author":[{"given":"RAYMOND","family":"HEMMECKE","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JASON","family":"MORTON","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ANNE","family":"SHIU","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"BERND","family":"STURMFELS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"OLIVER","family":"WIENAND","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2008,3,1]]},"reference":[{"key":"S0963548307008838_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548399003740"},{"key":"S0963548307008838_ref10","unstructured":"[10] Hemmecke R. , Takemura A. and Yoshida R. Computing holes in semi-groups. Preprint available at: math.CO\/0607599."},{"key":"S0963548307008838_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/S0246-0203(03)00020-7"},{"key":"S0963548307008838_ref11","unstructured":"[11] Hirai H. (2006) Sequences of stellar subdivisions. Preprint."},{"key":"S0963548307008838_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/009053606000000263"},{"key":"S0963548307008838_ref22","volume-title":"Lectures on Polytopes","author":"Ziegler","year":"1998"},{"key":"S0963548307008838_ref5","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1214\/aos\/1030563990","article-title":"Algebraic algorithms for sampling from conditional distributions","volume":"26","author":"Diaconis","year":"1998","journal-title":"Ann. Statist."},{"key":"S0963548307008838_ref19","doi-asserted-by":"publisher","DOI":"10.1080\/03081079308935207"},{"key":"S0963548307008838_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02766213"},{"key":"S0963548307008838_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8438-9_2"},{"key":"S0963548307008838_ref16","unstructured":"[16] Morton J. , Pachter L. , Shiu A. , Sturmfels B. and Wienand O. (2006) Geometry of rank tests. Probabilistic Graphical Models (PGM 3), Prague 2006. Available at: math.ST\/0605173."},{"key":"S0963548307008838_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00155-9"},{"key":"S0963548307008838_ref20","volume-title":"Probabilistic Conditional Independence Structures","author":"Studen\u00fd","year":"2005"},{"key":"S0963548307008838_ref2","unstructured":"[2] Aoki S. , Takemura A. and Yoshida R. (2005) Indispensable monomials of toric ideals and Markov bases. In Proc Asian Symposium on Computer Mathematics: ASCM 2005 (S. Pae and H. Park, eds), Korea Institute for Advanced Study, pp. 200\u2013202"},{"key":"S0963548307008838_ref15","volume-title":"Combinatorial Commutative Algebra","author":"Miller","year":"2004"},{"key":"S0963548307008838_ref21","volume-title":"Gr\u00f6bner Bases and Convex Polytopes","author":"Sturmfels","year":"1996"},{"key":"S0963548307008838_ref4","first-page":"59","article-title":"Computing the integral closure of an affine semigroup","volume":"39","author":"Bruns","year":"2001","journal-title":"Effective Methods in Algebraic and Analytic Geometry: Krak\u00f3w 2000. Univ. Iagel. Acta Math."},{"key":"S0963548307008838_ref1","unstructured":"[1] 4ti2: A software package for algebraic, geometric and combinatorial problems on linear spaces. Available at: http:\/\/www.4ti2.de."},{"key":"S0963548307008838_ref9","unstructured":"[9] Grayson D. and Stillman M. Macaulay2: A software system for research in algebraic geometry. Available at: http:\/\/www.math.uiuc.edu\/Macaulay2\/."},{"key":"S0963548307008838_ref6","volume-title":"Commutative Algebra with a View Toward Algebraic Geometry","author":"Eisenbud","year":"1995"},{"key":"S0963548307008838_ref17","unstructured":"[17] Postnikov A. (2005) Permutohedra, associahedra, and beyond. Preprint available at: math.CO\/0507163."},{"key":"S0963548307008838_ref18","unstructured":"[18] Postnikov A. , Reiner V. and Williams L. Faces of simple generalized permutohedra. Preprint available at: math.CO\/0609184."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548307008838","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,7]],"date-time":"2019-04-07T16:30:11Z","timestamp":1554654611000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548307008838\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,3]]}},"alternative-id":["S0963548307008838"],"URL":"https:\/\/doi.org\/10.1017\/s0963548307008838","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3]]}}}