{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T00:28:13Z","timestamp":1782692893587,"version":"3.54.5"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":9050,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,6]]},"abstract":"<jats:p>In this paper we study the varieties of Stone algebras (<jats:italic>S<\/jats:italic>, \u2227, \u2228, *, 0, 1) and double Stone algebras (<jats:italic>D<\/jats:italic>, \u2227, \u2228, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite \u2200\u2203-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].<\/jats:p><jats:p>Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also \u2135<jats:sub>0<\/jats:sub> categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.<\/jats:p><jats:p>This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra <jats:italic>A<\/jats:italic>, the universal Horn class <jats:bold>ISP<\/jats:bold><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002702X_inline1\"\/> has a model companion. Weispfenning [24], [25] discovered that this model companion is always \u2135<jats:sub>0<\/jats:sub> categorical and has a primitive recursive \u2200\u2203-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of <jats:bold>ISP<\/jats:bold><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002702X_inline1\"\/> is available. Burris and Werner [4] solve this problem in a special setting.<\/jats:p>","DOI":"10.2307\/2274853","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:30:21Z","timestamp":1146940221000},"page":"363-375","source":"Crossref","is-referenced-by-count":5,"title":["The structure of algebraically and existentially closed Stone and double Stone algebras"],"prefix":"10.1017","volume":"54","author":[{"given":"David M.","family":"Clark","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120002702X_ref025","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-16776-5_729"},{"key":"S002248120002702X_ref008","unstructured":"Day A. , Varieties of Heyting algebras. 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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":"S002248120002702X_ref003","first-page":"68","volume":"49","author":"Burris","year":"1984","journal-title":"Model companions for finitely generated universal Horn classes"},{"key":"S002248120002702X_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF02485733"},{"key":"S002248120002702X_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1979-0522263-8"},{"key":"S002248120002702X_ref017","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-24.3.507"},{"key":"S002248120002702X_ref018","unstructured":"Riedel H. , Existentially closed algebras and the Boolean product construction, Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1984."},{"key":"S002248120002702X_ref019","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19790253304"},{"key":"S002248120002702X_ref005","first-page":"3","article-title":"Topological quasivarieties","volume":"47","author":"Clark","year":"1984","journal-title":"Acta Universitatis Szegediensis. 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