{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T07:57:48Z","timestamp":1780387068770,"version":"3.54.1"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"7-8","license":[{"start":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T00:00:00Z","timestamp":1684540800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T00:00:00Z","timestamp":1684540800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Arch. Math. Logic"],"published-print":{"date-parts":[[2023,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory).<\/jats:p>","DOI":"10.1007\/s00153-023-00879-1","type":"journal-article","created":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T06:02:20Z","timestamp":1684562540000},"page":"1001-1032","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An AEC framework for fields with commuting automorphisms"],"prefix":"10.1007","volume":"62","author":[{"given":"Tapani","family":"Hyttinen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6967-3981","authenticated-orcid":false,"given":"Kaisa","family":"Kangas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,5,20]]},"reference":[{"issue":"11","key":"879_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2019.06.001","volume":"170","author":"N Ackerman","year":"2019","unstructured":"Ackerman, N., Boney, W., Vasey, S.: Categoricity in multiuniversal classes. Ann. Pure Appl. Logic 170(11), 102712 (2019)","journal-title":"Ann. Pure Appl. Logic"},{"key":"879_CR2","unstructured":"Baldwin, J.: Categoricity. University Lecture Series, vol. 50. AMS, Providence (2009). http:\/\/homepages.math.uic.edu\/~jbaldwin\/pub\/AEClec.pdf"},{"key":"879_CR3","doi-asserted-by":"publisher","DOI":"10.1017\/9781316717035","volume-title":"Fundamentals of Stability Theory","author":"J Baldwin","year":"2017","unstructured":"Baldwin, J.: Fundamentals of Stability Theory. Cambridge University Press (2017)"},{"key":"879_CR4","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1112\/plms.12180","volume":"118","author":"\u00d6 Beyarslan","year":"2019","unstructured":"Beyarslan, \u00d6., Kowalski, P.: Model theory of fields with virtually free group actions. Proc. Lond. Math. Soc. (3) 118, 221\u2013256 (2019)","journal-title":"Proc. Lond. Math. Soc. (3)"},{"key":"879_CR5","unstructured":"Beyarslan, \u00d6., Kowalski, P.: Galois actions of finitely generated groups rarely have model companions. Preprint. https:\/\/arxiv.org\/pdf\/2210.00800.pdf"},{"issue":"1","key":"879_CR6","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1090\/S0894-0347-02-00407-1","volume":"16","author":"S Buechler","year":"2002","unstructured":"Buechler, S., Lessmann, O.: Simple homogeneous models. J. Am. Math. Soc. 16(1), 91\u2013121 (2002)","journal-title":"J. Am. Math. Soc."},{"key":"879_CR7","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1017\/CBO9780511565670.004","volume-title":"Models and Computability","author":"Z Chatzidakis","year":"1999","unstructured":"Chatzidakis, Z.: Simplicity and independence for pseudo-algebraically closed fields. In: Cooper, B., Truss, J. (eds.) Models and Computability, pp. 41\u201361. Cambridge University Press (1999)"},{"issue":"8","key":"879_CR8","doi-asserted-by":"publisher","first-page":"2997","DOI":"10.1090\/S0002-9947-99-02498-8","volume":"351","author":"Z Chatzidakis","year":"1999","unstructured":"Chatzidakis, Z., Hrushovski, E.: Model theory of difference fields. Trans. Am. Math. Soc. 351(8), 2997\u20133071 (1999)","journal-title":"Trans. Am. Math. Soc."},{"issue":"4","key":"879_CR9","first-page":"653","volume":"7","author":"Z Chatzidakis","year":"2008","unstructured":"Chatzidakis, Z., Hrushovski, E.: Difference fields and descent in algebraic dynamics. I. J. IMJ 7(4), 653\u2013686 (2008)","journal-title":"J. IMJ"},{"issue":"4","key":"879_CR10","first-page":"687","volume":"7","author":"Z Chatzidakis","year":"2008","unstructured":"Chatzidakis, Z., Hrushovski, E.: Difference fields and descent in algebraic dynamics. II. J. IMJ 7(4), 687\u2013704 (2008)","journal-title":"J. IMJ"},{"key":"879_CR11","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1112\/S0024611502013576","volume":"85","author":"Z Chatzidakis","year":"2002","unstructured":"Chatzidakis, Z., Hrushovski, E., Peterzil, Y.: Model theory of difference fields II: periodic ideals and the trichotomy in all characteristics. Proc. Lond. Math. Soc. (3) 85, 257\u2013311 (2002)","journal-title":"Proc. Lond. Math. Soc. (3)"},{"key":"879_CR12","volume-title":"Basic Algebra. Groups, Rings and Fields","author":"PM Cohn","year":"2003","unstructured":"Cohn, P.M.: Basic Algebra. Groups, Rings and Fields. Springer (2003)"},{"key":"879_CR13","volume-title":"Difference Algebra, Tracts in Mathematics","author":"RM Cohn","year":"1965","unstructured":"Cohn, R.M.: Difference Algebra, Tracts in Mathematics, vol. 17. Interscience Pub. (1965)"},{"issue":"1","key":"879_CR14","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1080\/00029890.1981.11995181","volume":"88","author":"R Dean","year":"1981","unstructured":"Dean, R.: A rational polynomial whose group is the quaternions. Am. Math. Mon. 88(1), 42\u201345 (1981)","journal-title":"Am. Math. Mon."},{"key":"879_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-07216-5","volume-title":"Field Arithmetic","author":"MD Fried","year":"1986","unstructured":"Fried, M.D., Jarden, M.: Field Arithmetic. Springer (1986)"},{"issue":"1","key":"879_CR16","doi-asserted-by":"publisher","first-page":"1850003","DOI":"10.1142\/S0219061318500034","volume":"18","author":"D Hoffmann","year":"2018","unstructured":"Hoffmann, D., Kowalski, P.: Existentially closed fields with finite group actions. J. Math. Logic 18(1), 1850003 (2018)","journal-title":"J. Math. Logic"},{"key":"879_CR17","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/S0168-0072(01)00096-3","volume":"112","author":"E Hrushovski","year":"2001","unstructured":"Hrushovski, E.: The Manin\u2013Mumford conjecture and the model theory of difference fields. Ann. Pure Appl. Logic 112, 43\u2013115 (2001)","journal-title":"Ann. Pure Appl. Logic"},{"key":"879_CR18","unstructured":"Kikyo, H.: On generic predicates and automorphisms. RIMS Kokyuroku 1390 (2004)"},{"key":"879_CR19","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/S0168-0072(97)00019-5","volume":"88","author":"P Kim","year":"1997","unstructured":"Kim, P., Pillay, A.: Simple theories. Ann. Pure Appl. Logic 88, 149\u2013164 (1997)","journal-title":"Ann. Pure Appl. Logic"},{"key":"879_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-6947-5","volume-title":"Difference Algebra","author":"A Levin","year":"2008","unstructured":"Levin, A.: Difference Algebra. Springer (2008)"},{"key":"879_CR21","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/S0168-0072(97)00020-1","volume":"88","author":"A Macintyre","year":"1997","unstructured":"Macintyre, A.: Generic automorphisms of fields. Ann. Pure Appl. Logic 88, 165\u2013180 (1997)","journal-title":"Ann. Pure Appl. Logic"},{"key":"879_CR22","volume-title":"Model Theory: An Introduction","author":"D Marker","year":"2000","unstructured":"Marker, D.: Model Theory: An Introduction. Springer (2000)"},{"key":"879_CR23","volume-title":"Forking in the Category of Existentially Closed Structures, Connections Between Model Theory ands Algebraic and Analytic Geometry, Quaderni di Matematica","author":"A Pillay","year":"2000","unstructured":"Pillay, A.: Forking in the Category of Existentially Closed Structures, Connections Between Model Theory ands Algebraic and Analytic Geometry, Quaderni di Matematica, vol. 6. University of Naples (2000)"},{"key":"879_CR24","volume-title":"Classification Theory and the Number of Nonisomorphic Models","author":"S Shelah","year":"1978","unstructured":"Shelah, S.: Classification Theory and the Number of Nonisomorphic Models. North-Holland, Amsterdam (1978)"},{"issue":"3","key":"879_CR25","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0003-4843(80)90009-1","volume":"19","author":"S Shelah","year":"1980","unstructured":"Shelah, S.: Simple unstable theories. Ann. Math. Logic 19(3), 177\u2013203 (1980)","journal-title":"Ann. Math. Logic"},{"key":"879_CR26","unstructured":"Sj\u00f6gren, N.: The Model Theory of Fields with a Group Action. Research Reports in Mathematics. Department of Mathematics, Stockholm University (2005). https:\/\/www2.math.su.se\/reports\/2005\/7\/"},{"key":"879_CR27","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/s00153-016-0481-z","volume":"55","author":"S Vasey","year":"2016","unstructured":"Vasey, S.: Infinitary stability theory. Arch. Math. Logic 55, 567\u2013592 (2016)","journal-title":"Arch. Math. Logic"},{"issue":"9","key":"879_CR28","doi-asserted-by":"publisher","first-page":"1609","DOI":"10.1016\/j.apal.2017.03.003","volume":"168","author":"S Vasey","year":"2017","unstructured":"Vasey, S.: Shelah\u2019s eventual categoricity conjecture in universal classes: part I. Ann. Pure Appl. Logic 168(9), 1609\u20131642 (2017)","journal-title":"Ann. Pure Appl. Logic"}],"container-title":["Archive for Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-023-00879-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00153-023-00879-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-023-00879-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,20]],"date-time":"2023-09-20T03:27:51Z","timestamp":1695180471000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00153-023-00879-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,20]]},"references-count":28,"journal-issue":{"issue":"7-8","published-print":{"date-parts":[[2023,11]]}},"alternative-id":["879"],"URL":"https:\/\/doi.org\/10.1007\/s00153-023-00879-1","relation":{},"ISSN":["0933-5846","1432-0665"],"issn-type":[{"value":"0933-5846","type":"print"},{"value":"1432-0665","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,20]]},"assertion":[{"value":"29 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}