{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T22:04:27Z","timestamp":1747173867340,"version":"3.40.5"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T00:00:00Z","timestamp":1662940800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove a strengthened version of Shavrukov\u2019s result on the non-isomorphism of diagonalizable algebras of two <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000302_inline1.png\"\/><jats:tex-math>\n$\\Sigma _1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000302_inline2.png\"\/><jats:tex-math>\n$(\\mathfrak {L}_T, \\Box _T\\Box _T)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> onto the algebra <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000302_inline3.png\"\/><jats:tex-math>\n$(\\mathfrak {L}_T, \\Box _T)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The case of bimodal diagonalizable algebras is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras.<\/jats:p>","DOI":"10.1017\/s1755020322000302","type":"journal-article","created":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T04:54:48Z","timestamp":1662958488000},"page":"206-243","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["ON SHAVRUKOV\u2019S NON-ISOMORPHISM THEOREM FOR DIAGONALIZABLE ALGEBRAS"],"prefix":"10.1017","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6526-9719","authenticated-orcid":false,"given":"EVGENY A.","family":"KOLMAKOV","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,9,12]]},"reference":[{"key":"S1755020322000302_r11","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2018.82"},{"key":"S1755020322000302_r3","first-page":"317","volume-title":"Logic and Algebra","author":"Beklemishev","year":"1996"},{"key":"S1755020322000302_r13","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061308000774"},{"key":"S1755020322000302_r15","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19720180405"},{"key":"S1755020322000302_r17","first-page":"312","volume-title":"Model Theory of Algebra and Arithmetic. 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(2011). Diagonalizable algebras and the length of proofs. 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