{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T12:13:24Z","timestamp":1785413604742,"version":"3.56.0"},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat Ab as the base system and refer to its expansions as\n                    <jats:italic>superabelian logics<\/jats:italic>\n                    . The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S1755020326101129_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">2 Superscript 2 Super Superscript omega<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <jats:tex-math>$2^{2^\\omega }$<\/jats:tex-math>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>\u03c9<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    distinct logics in this family. Second, we introduce\n                    <jats:italic>pointed Abelian logic<\/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\" content-type=\"simple\" xlink:href=\"S1755020326101129_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">pAb<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <jats:tex-math>${\\text {pAb}}$<\/jats:tex-math>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mtext>pAb<\/mml:mtext>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes\n                    <jats:italic>\u0141ukasiewicz unbound logic<\/jats:italic>\n                    . We provide axiomatizations for its finitary and infinitary versions as extensions of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S1755020326101129_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">pAb<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <jats:tex-math>${\\text {pAb}}$<\/jats:tex-math>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mtext>pAb<\/mml:mtext>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and establish their precise relationship with standard \u0141ukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.\n                  <\/jats:p>","DOI":"10.1017\/s1755020326101129","type":"journal-article","created":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T08:33:56Z","timestamp":1774341236000},"page":"218-244","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["SUPERABELIAN LOGICS"],"prefix":"10.1017","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3617-1392","authenticated-orcid":false,"given":"PETR","family":"CINTULA","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0496n6574","id-type":"ROR","asserted-by":"publisher"}],"name":"INSTITUTE OF COMPUTER SCIENCE CZECH ACADEMY OF SCIENCES"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2746-0042","authenticated-orcid":false,"given":"FILIP","family":"JANKOVEC","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0496n6574","id-type":"ROR","asserted-by":"publisher"}],"name":"INSTITUTE OF COMPUTER SCIENCE CZECH ACADEMY OF SCIENCES"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4910-599X","authenticated-orcid":false,"given":"CARLES","family":"NOGUERA","sequence":"additional","affiliation":[{"name":"UNIVERSITY OF SIENA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"S1755020326101129_r10","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/176"},{"key":"S1755020326101129_r9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-85675-5"},{"key":"S1755020326101129_r5","first-page":"161","volume-title":"Logic Colloquium \u201988","volume":"127","author":"Casari","year":"1989"},{"key":"S1755020326101129_r22","first-page":"71","article-title":"Universalnaya teoriya strukturno uporyadochennykh abelevykh grupp (universal theory of lattice-ordered Abelian groups)","volume":"5","author":"Khisamiev","year":"1966","journal-title":"Algebra Logika"},{"key":"S1755020326101129_r16","volume-title":"Algebraic Theory of Quasivarieties","author":"Gorbunov","year":"1998"},{"key":"S1755020326101129_r20","first-page":"1","article-title":"Die Axiome der Quantit\u00e4t und die Lehre vom Mass","volume":"53","author":"H\u00f6lder","year":"1901","journal-title":"Berichte \u00fcber die Verhandlungen der K\u00f6niglich S\u00e4chsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse"},{"key":"S1755020326101129_r7","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-006-0011-5"},{"key":"S1755020326101129_r13","volume-title":"Partially Ordered Algebraic Systems","author":"Fuchs","year":"1963"},{"key":"S1755020326101129_r19","doi-asserted-by":"publisher","DOI":"10.2307\/2271339"},{"key":"S1755020326101129_r1","volume-title":"Universal Algebra: Fundamentals and Selected Topics","volume":"301","author":"Bergman","year":"2012"},{"key":"S1755020326101129_r21","unstructured":"[21] Jankovec, F. 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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"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}