{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T10:59:11Z","timestamp":1770461951715,"version":"3.49.0"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T00:00:00Z","timestamp":1701820800000},"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":["J. symb. log."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We generalise the properties\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline1.png\"\/>\n                        <jats:tex-math>$\\mathsf {OP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline2.png\"\/>\n                        <jats:tex-math>$\\mathsf {IP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:italic>k<\/jats:italic>\n                    -\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline3.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline4.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}_{1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:italic>k<\/jats:italic>\n                    -\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline5.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}_{2}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline6.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}_{1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline7.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}_{2}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline8.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}_{3}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline9.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory\n                    <jats:italic>T<\/jats:italic>\n                    has\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline10.png\"\/>\n                        <jats:tex-math>$\\mathsf {OP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    iff it has\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline11.png\"\/>\n                        <jats:tex-math>$\\mathsf {IP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    or\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline12.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}_{1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and that\n                    <jats:italic>T<\/jats:italic>\n                    has\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline13.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    iff it has\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline14.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}_{1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    or\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline15.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}_{2}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , analogous to the well-known results in full first-order logic where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline16.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}_{1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is replaced by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline17.png\"\/>\n                        <jats:tex-math>$\\mathsf {SOP}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in the former and by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481223000890_inline18.png\"\/>\n                        <jats:tex-math>$\\mathsf {TP}_{1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.\n                  <\/jats:p>","DOI":"10.1017\/jsl.2023.89","type":"journal-article","created":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T03:21:45Z","timestamp":1701832905000},"page":"1639-1663","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["DIVIDING LINES BETWEEN POSITIVE THEORIES"],"prefix":"10.1017","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7551-6122","authenticated-orcid":false,"given":"ANNA","family":"DMITRIEVA","sequence":"first","affiliation":[{"name":"UNIVERSITY OF EAST ANGLIA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9171-8522","authenticated-orcid":false,"given":"FRANCESCO","family":"GALLINARO","sequence":"additional","affiliation":[{"name":"MATHEMATISCHES INSTITUT, ALBERT-LUDWIGS-UNIVERSIT\u00c4T FREIBURG"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0419-7061","authenticated-orcid":false,"given":"MARK","family":"KAMSMA","sequence":"additional","affiliation":[{"name":"IMPERIAL COLLEGE LONDON"}]}],"member":"56","published-online":{"date-parts":[[2023,12,6]]},"reference":[{"key":"S0022481223000890_r9","doi-asserted-by":"publisher","DOI":"10.1007\/s001530200000"},{"key":"S0022481223000890_r19","volume-title":"Classification Theory and the Number of Nonisomorphic Models","author":"Shelah","year":"1990"},{"key":"S0022481223000890_r20","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00066-6"},{"key":"S0022481223000890_r13","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-013-0363-6"},{"key":"S0022481223000890_r3","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061303000297"},{"key":"S0022481223000890_r2","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061303000212"},{"key":"S0022481223000890_r11","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2023.103268"},{"key":"S0022481223000890_r7","doi-asserted-by":"publisher","DOI":"10.2140\/mt.2022.1.55"},{"key":"S0022481223000890_r18","first-page":"241","article-title":"The lazy model-theoretician\u2019s guide to stability","volume":"18","author":"Shelah","year":"1975","journal-title":"Logique et Analyse"},{"key":"S0022481223000890_r1","unstructured":"[1] Belkasmi, M. , Contributions \u00e0 la th\u00e9orie des mod\u00e8les Positive , Ph.D. thesis, Universit\u00e9 Claude Bernard Lyon 1, Lyon, 2012."},{"key":"S0022481223000890_r6","unstructured":"[6] Conant, G. , Dividing lines in unstable theories. Manuscript. 2012."},{"key":"S0022481223000890_r5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735219.011"},{"key":"S0022481223000890_r12","unstructured":"[12] Kamsma, M. , Positive indiscernibles , preprint, 2023, arXiv:2305.14127."},{"key":"S0022481223000890_r15","first-page":"23","article-title":"Forking in the category of existentially closed structures","volume":"6","author":"Pillay","year":"2000","journal-title":"Quaderni di Matematica"},{"key":"S0022481223000890_r4","doi-asserted-by":"publisher","DOI":"10.4064\/fm179-3-2"},{"key":"S0022481223000890_r14","unstructured":"[14] Mutchnik, S. , On NSOP2 theories, preprint, 2022, arXiv:2206.08512."},{"key":"S0022481223000890_r10","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-021-2089-1"},{"key":"S0022481223000890_r16","doi-asserted-by":"publisher","DOI":"10.1007\/s11787-018-0185-8"},{"key":"S0022481223000890_r21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139015417"},{"key":"S0022481223000890_r17","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90007-0"},{"key":"S0022481223000890_r8","doi-asserted-by":"crossref","unstructured":"[8] Dobrowolski, J. and Mennuni, R. , The amalgamation property for automorphisms of ordered abelian groups, preprint, 2023, arXiv:2209.03944.","DOI":"10.1090\/tran\/9217"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481223000890","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T06:57:59Z","timestamp":1770361079000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481223000890\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,6]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["S0022481223000890"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2023.89","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,6]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. 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