{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T08:36:37Z","timestamp":1774946197213,"version":"3.50.1"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"unspecified","delay-in-days":14,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Bull. symb. log"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We say that a computable structure\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline1.png\"\/>\n                        <jats:tex-math>$\\mathcal {A}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is computably categorical if for every computable copy\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline2.png\"\/>\n                        <jats:tex-math>$\\mathcal {B}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , there exists a computable isomorphism\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline3.png\"\/>\n                        <jats:tex-math>$f:\\mathcal {A}\\to \\mathcal {B}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . This notion can be relativized to a degree\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline4.png\"\/>\n                        <jats:tex-math>$\\mathbf {d}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    by saying that a computable structure\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline5.png\"\/>\n                        <jats:tex-math>$\\mathcal {A}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is computably categorical relative to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline6.png\"\/>\n                        <jats:tex-math>$\\mathbf {d}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    if for every\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline7.png\"\/>\n                        <jats:tex-math>$\\mathbf {d}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -computable copy\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline8.png\"\/>\n                        <jats:tex-math>$\\mathcal {B}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline9.png\"\/>\n                        <jats:tex-math>$\\mathcal {A}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , there exists a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline10.png\"\/>\n                        <jats:tex-math>$\\mathbf {d}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -computable isomorphism\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline11.png\"\/>\n                        <jats:tex-math>$f:\\mathcal {A}\\to \\mathcal {B}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . A\u00a0key part of this thesis is to study the behavior of this notion of categoricity in the computably enumerable degrees.\n                  <\/jats:p>\n                  <jats:p>\n                    The main theorem in Chapter\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline12.png\"\/>\n                        <jats:tex-math>$1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    states that given any computable partially ordered set\n                    <jats:italic>P<\/jats:italic>\n                    and any computable partition\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline13.png\"\/>\n                        <jats:tex-math>$P=P_0\\sqcup P_1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , there exists an embedding\n                    <jats:italic>h<\/jats:italic>\n                    of\n                    <jats:italic>P<\/jats:italic>\n                    into the c.e. degrees and a computable graph\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline14.png\"\/>\n                        <jats:tex-math>$\\mathcal {G}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    which is computably categorical, computably categorical relative to all degrees in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline15.png\"\/>\n                        <jats:tex-math>$h(P_0)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and is not computably categorical relative to any degree in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline16.png\"\/>\n                        <jats:tex-math>$h(P_1)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We also show that by using largely the same techniques alongside a standard construction of minimal pairs, we can embed a four-element diamond lattice into the c.e. degrees in the style of the main result of Chapter\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline17.png\"\/>\n                        <jats:tex-math>$1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    We then apply some of the techniques used in this thesis to study the behavior of this notion in the context of generic degrees in Chapter\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline18.png\"\/>\n                        <jats:tex-math>$2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Additionally, we show that several classes of structures admit a computable example that witnesses the pathological behavior of categoricity relative to a degree as seen in Chapter\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline19.png\"\/>\n                        <jats:tex-math>$1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    \u2019s main theorem.\n                  <\/jats:p>\n                  <jats:p>\n                    Lastly, in the context of reverse mathematics, we investigate the reverse mathematical strength of a topological principle named\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline20.png\"\/>\n                        <jats:tex-math>$\\mathsf {wGS}^{\\operatorname {cl}}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , a weakened version of the Ginsburg\u2013Sands theorem which states that every infinite topological space contains one of the following five topologies as a subspace, with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100929_inline21.png\"\/>\n                        <jats:tex-math>$\\mathbb {N}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    as the underlying set: discrete, indiscrete, cofinite, initial segment, or final segment.\n                  <\/jats:p>\n                  <jats:p>Abstract prepared by Java Darleen Villano<\/jats:p>\n                  <jats:p>\n                    <jats:italic>E-mail<\/jats:italic>\n                    :\n                    <jats:email>java.villano@utoronto.ca<\/jats:email>\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:italic>URL<\/jats:italic>\n                    :\n                    <jats:uri xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/javavillano.github.io\/thesisfinal.pdf\">https:\/\/javavillano.github.io\/thesisfinal.pdf<\/jats:uri>\n                  <\/jats:p>","DOI":"10.1017\/bsl.2025.10092","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:59:23Z","timestamp":1765760363000},"page":"694-694","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["J\n                    <scp>ava<\/scp>\n                    D\n                    <scp>arleen<\/scp>\n                    V\n                    <scp>illano<\/scp>\n                    .\n                    <i>\n                      <b>Computable Categoricity, and Topology in Reverse Mathematics<\/b>\n                    <\/i>\n                    . University of Connecticut. 2025. Supervised by Reed Solomon and Damir Dzhafarov. MSC: 03C57, 03B30."],"prefix":"10.1017","volume":"31","member":"56","published-online":{"date-parts":[[2025,12,15]]},"container-title":["The Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898625100929","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T07:26:47Z","timestamp":1774942007000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898625100929\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["S1079898625100929"],"URL":"https:\/\/doi.org\/10.1017\/bsl.2025.10092","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}