{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T21:34:19Z","timestamp":1775511259791,"version":"3.50.1"},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12429,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:p>In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings.<\/jats:p><jats:p>The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering \u3008<jats:italic>A, R<\/jats:italic>\u3009 to be <jats:italic>n<\/jats:italic>-dimensional if there are <jats:italic>n<\/jats:italic> linear orderings of <jats:italic>A<\/jats:italic>, \u3008<jats:italic>A, L<\/jats:italic><jats:sub>1<\/jats:sub>\u3009, \u3008<jats:italic>A, L<\/jats:italic><jats:sub>2<\/jats:sub>\u3009 \u2026, \u3008<jats:italic>A, L<jats:sub>n<\/jats:sub><\/jats:italic>\u3009 such that <jats:italic>R<\/jats:italic> = <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub> \u2229 <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub> \u2229 \u2026 \u2229 <jats:italic>L<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>. Thus, for example, if <jats:italic>Q<\/jats:italic> is the linear ordering of the rationals, then the (rational) plane <jats:italic>Q<\/jats:italic> \u00d7 <jats:italic>Q<\/jats:italic> with the product ordering (\u3008<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub>\u3009 \u2264<jats:sub><jats:italic>Q<\/jats:italic>\u00d7<jats:italic>Q<\/jats:italic><\/jats:sub> \u3008<jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>y<\/jats:italic><jats:sub>2<\/jats:sub>, if and only if <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub> \u2264 <jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub> and <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub> \u2264 <jats:italic>y<\/jats:italic><jats:sub>2<\/jats:sub>) is 2-dimensional, since \u2264<jats:sub><jats:italic>Q<\/jats:italic>\u00d7<jats:italic>Q<\/jats:italic><\/jats:sub> is the intersection of the two lexicographic orderings of <jats:italic>Q<\/jats:italic> \u00d7 <jats:italic>Q<\/jats:italic>. In fact, as shown by Dushnik and Miller, a countable partial ordering is <jats:italic>n<\/jats:italic>-dimensional if and only if it can be embedded as a subordering of <jats:italic>Q<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>.<\/jats:p><jats:p>Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes <jats:italic>explicitly<\/jats:italic> a collection <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047204_inline1\"\/> of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047204_inline1\"\/> as a subordering.<\/jats:p>","DOI":"10.2307\/2273359","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:51:56Z","timestamp":1146952316000},"page":"121-132","source":"Crossref","is-referenced-by-count":8,"title":["Two-dimensional partial orderings: Recursive model theory"],"prefix":"10.1017","volume":"45","author":[{"given":"Alfred B.","family":"Manaster","sequence":"first","affiliation":[]},{"given":"Joseph G.","family":"Rosenstein","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200047204_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(76)80011-8"},{"key":"S0022481200047204_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/2371374"},{"key":"S0022481200047204_ref009","first-page":"133","volume":"45","author":"Manaster","year":"1980","journal-title":"Two-dimensional partial orderings; Undecidability"},{"key":"S0022481200047204_ref005","first-page":"268","volume":"37","author":"Jockusch","year":"1972","journal-title":"Ramsey's theorem and recursion theory"},{"key":"S0022481200047204_ref002","volume-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete","volume":"81","author":"Crossley","year":"1974"},{"key":"S0022481200047204_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/mana.19700460115"},{"key":"S0022481200047204_ref007","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1977-040-3"},{"key":"S0022481200047204_ref001","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230020103"},{"key":"S0022481200047204_ref006","first-page":"33","article-title":"classes and degrees of theories","volume":"173","author":"Jockusch","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200047204_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-25.4.615"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200047204","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T19:36:10Z","timestamp":1558899370000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200047204\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1980,3]]}},"alternative-id":["S0022481200047204"],"URL":"https:\/\/doi.org\/10.2307\/2273359","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,3]]}}}