{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T10:18:39Z","timestamp":1693390719239},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11515,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,9]]},"abstract":"<jats:p>Let <jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, 0 \u2264 <jats:italic>n<\/jats:italic> \u2264 <jats:italic>\u03c9<\/jats:italic>, be the equational classes of distributive <jats:italic>p<\/jats:italic>-algebras (precise definitions are given in \u00a71). It has been known for some time that the elementary theories <jats:italic>T<\/jats:italic><jats:sub>n<\/jats:sub> of <jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> possess model companions <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline01\" \/>; see, e.g., [6] and [14] and the references given there. However, no axiomatizations of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline01\" \/> were given, with the exception of <jats:italic>n<\/jats:italic> = 0 (Boolean case) and <jats:italic>n<\/jats:italic>= 1 (Stonian case). While the first case belongs to the folklore of the subject (see [6], also [11]), the second case presented considerable difficulties (see Schmitt [13]). Schmitt's use of methods characteristic for Stone algebras seems to prevent a ready adaptation of his results to the cases <jats:italic>n<\/jats:italic> \u2265 2.<\/jats:p><jats:p>The natural way to get a hold on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline01\" \/> is to determine the class <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) of existentially complete members of <jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>: Since <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline01\" \/> exists, it equals the elementary theory of <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>). The present author succeeded [12] in solving the simpler problem of determining the classes <jats:italic>A<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) of algebraically closed algebras in <jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> (exact definitions of <jats:italic>A<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) and <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) are given in \u00a71) for all 0 &gt; <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>. A(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) is easier to handle since it contains sufficiently many \u201csmall\u201d algebras-viz. finite direct products of certain subdirectly irreducibles-in terms of which the members of <jats:italic>A<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) may be analyzed (in contrast, all members of <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) are infinite and \u2135-homogeneous). As it turns out, <jats:italic>A<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) is finitely axiomatizable for all <jats:italic>n<\/jats:italic>, and comparing the theories of <jats:italic>A<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub>0<\/jats:sub>), A(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub>1<\/jats:sub>) with the explicitly known theories of <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub>0<\/jats:sub>), <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub>1<\/jats:sub>)-viz. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline02\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline03\" \/>, a reasonable conjecture for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline01\" \/>, 2 \u2264 <jats:italic>n<\/jats:italic> \u2264 <jats:italic>\u03c9<\/jats:italic>, is immediate. The main part of this paper is concerned with verifying that the conditions formalized by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044042_inline01\" \/> suffice to describe the algebras in <jats:italic>E<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) (necessity is easy). This verification rests on the same combinatorial techniques as used in [12] to describe the members of <jats:italic>A<\/jats:italic>(<jats:underline><jats:italic>B<\/jats:italic><\/jats:underline><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>).<\/jats:p>","DOI":"10.2307\/2273597","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:00:58Z","timestamp":1146952858000},"page":"680-688","source":"Crossref","is-referenced-by-count":9,"title":["Model companions of distributive <i>p<\/i>-algebras"],"prefix":"10.1017","volume":"47","author":[{"given":"J\u00fcrg","family":"Schmid","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044042_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF02011865"},{"key":"S0022481200044042_ref013","first-page":"135","volume-title":"Annales Scientifiques de VUniver-sit\u00e9 de Clermont, S\u00e9rie Math\u00e9matique","author":"Schmitt","year":"1976"},{"key":"S0022481200044042_ref012","unstructured":"[12] Schmid J. , Algebraically closed distributive p-algebras, Algebra Universalis (to appear)."},{"key":"S0022481200044042_ref011","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19790253304"},{"key":"S0022481200044042_ref010","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1970-101-4"},{"key":"S0022481200044042_ref008","doi-asserted-by":"publisher","DOI":"10.1002\/mana.19720530109"},{"key":"S0022481200044042_ref006","volume-title":"Lecture Notes in Mathematics","volume":"454","author":"Hirschfeld","year":"1975"},{"key":"S0022481200044042_ref005","first-page":"475","article-title":"The structure of pseudocomplemented distributive lattices. 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II: Congruence extension and amalgamation","volume":"156","author":"Gr\u00e4tzer","year":"1971","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200044042_ref001","first-page":"33","volume-title":"Proceedings of the Ulm Lattice Theory Conference","author":"Burris","year":"1975"},{"key":"S0022481200044042_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90016-7"},{"key":"S0022481200044042_ref003","volume-title":"Lattice theory: First concepts and distributive lattices","author":"Gr\u00e4tzer","year":"1971"},{"key":"S0022481200044042_ref009","first-page":"160","article-title":"Die Kennzeichnung der distributiven pseudokomplement\u00e4ren Halbverb\u00e4nde","volume":"241","author":"Katri\u0148\u00e1k","year":"1970","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044042","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T20:47:13Z","timestamp":1558730833000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044042\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,9]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1982,9]]}},"alternative-id":["S0022481200044042"],"URL":"https:\/\/doi.org\/10.2307\/2273597","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,9]]}}}