{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T12:36:46Z","timestamp":1770813406561,"version":"3.50.1"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004004","name":"Universit\u00e0 degli Studi di Trento","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004004","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Arch. Math. Logic"],"published-print":{"date-parts":[[2026,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We study model-theoretic properties of a logic whose formulas take values in suitable Riesz spaces. In addition to having a set of truth values which is not linearly ordered, there is no absolute truth or falsehood value. We take inspiration from the version of continuous logic developed in [8] by Ben-Yaacov\n                    <jats:italic>et al.<\/jats:italic>\n                    and from the more general approach recently proposed in [11] by Keisler. We extend to our framework a number of results obtained by the above mentioned authors. Under suitable assumptions on the underlying Riesz space, we provide an ultraproduct construction and prove a \u0141o\u015b theorem, from which compactness follows. In the framework of Riesz spaces we also address definability issues and we extend metric notions, obtaining analogues of Keisler\u2019s pre-metric structures and pre-metric expansions of theories defined in Keisler. In particular, we prove that every theory in a countable language has a pre-metric expansion with a so-called pseudometric approximate distance.\n                  <\/jats:p>","DOI":"10.1007\/s00153-025-00990-5","type":"journal-article","created":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T09:22:24Z","timestamp":1762766544000},"page":"239-273","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Results in model theory for Riesz-valued structures"],"prefix":"10.1007","volume":"65","author":[{"given":"Stefano","family":"Baratella","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,11,8]]},"reference":[{"key":"990_CR1","volume-title":"Problems in operator theory, Graduate studies in Mathematics","author":"YA Abramovich","year":"2000","unstructured":"Abramovich, Y.A., Aliprantis, C.D.: Problems in operator theory, Graduate studies in Mathematics, vol. 51. American Mathematical Society, Providence (2000)"},{"key":"990_CR2","volume-title":"Locally solid Riesz spaces","author":"CD Aliprantis","year":"1978","unstructured":"Aliprantis, C.D., Burkinshaw, O.: Locally solid Riesz spaces. Academic Press, New York (1978)"},{"key":"990_CR3","doi-asserted-by":"publisher","first-page":"375","DOI":"10.12775\/TMNA.2009.048","volume":"34","author":"I Altun","year":"2009","unstructured":"Altun, I., \u00c7evik, C.: Vector metric spaces and some properties. Topological methods in nonlinear analysis 34, 375\u2013382 (2009)","journal-title":"Topological methods in nonlinear analysis"},{"key":"990_CR4","unstructured":"Albert, J.-M., Hart, B.: Metric logical categories and conceptual completeness for first order continuous logic. arXiv:1067.03068v1 (2016)"},{"key":"990_CR5","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1007\/s00153-017-0558-3","volume":"56","author":"S Baratella","year":"2017","unstructured":"Baratella, S.: A predicate extension of real valued logic. Arch. Math. Logic 56, 585\u2013605 (2017)","journal-title":"Arch. Math. Logic"},{"issue":"1\u20132","key":"990_CR6","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1002\/malq.201300068","volume":"61","author":"S Baratella","year":"2015","unstructured":"Baratella, S., Zambella, D.: The $${\\mathbb{R} }$$eal truth. Math. Log. Q. 61(1\u20132), 32\u201344 (2015)","journal-title":"Math. Log. Q."},{"key":"990_CR7","volume-title":"Models and ultraproducts-an introduction","author":"JL Bell","year":"2006","unstructured":"Bell, J.L., Slomson, A.B.: Models and ultraproducts-an introduction. Dover, New York (2006)"},{"key":"990_CR8","first-page":"315","volume":"350","author":"I Ben-Yaacov","year":"2008","unstructured":"Ben-Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. In Model Theory with Applications to Algebra and Analysis. vol. 2, London Math. Society Lecture Note Series 350, 315\u2013427 (2008)","journal-title":"vol. 2, London Math. Society Lecture Note Series"},{"issue":"4","key":"990_CR9","first-page":"759","volume":"21","author":"O Burkinshaw","year":"1977","unstructured":"Burkinshaw, O., Doods, P.: Disjoint sequences, compactness and semi-reflexivity in locally convex Riesz spaces. Ill. J. Math. 21(4), 759\u2013775 (1977)","journal-title":"Ill. J. Math."},{"issue":"4","key":"990_CR10","doi-asserted-by":"publisher","first-page":"1909","DOI":"10.2307\/2694984","volume":"66","author":"J Iovino","year":"2001","unstructured":"Iovino, J.: On the maximality of logics with approximations. J. Symb. Log. 66(4), 1909\u20131918 (2001)","journal-title":"J. Symb. Log."},{"key":"990_CR11","unstructured":"Keisler, H.J.: Model theory for real valued structures, arXiv:2005.11851v2 [math.LO]"},{"key":"990_CR12","volume-title":"Riesz spaces","author":"WAJ Luxemburg","year":"1971","unstructured":"Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces, vol. 1. North-Holland, Amsterdam (1971)"}],"container-title":["Archive for Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-025-00990-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00153-025-00990-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-025-00990-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T17:23:54Z","timestamp":1770398634000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00153-025-00990-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,8]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["990"],"URL":"https:\/\/doi.org\/10.1007\/s00153-025-00990-5","relation":{},"ISSN":["0933-5846","1432-0665"],"issn-type":[{"value":"0933-5846","type":"print"},{"value":"1432-0665","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,8]]},"assertion":[{"value":"28 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 October 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}