{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T00:25:09Z","timestamp":1759883109666,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Hybrid sets are defined as multisets having also negative multiplicities, i.e. as functions from a crisp set to the group of all integers. In this article, we introduce a significant advancement in hybrid sets through the concept of group-valued multisets. These multisets map elements of a set X to an arbitrary group, ensuring that each multiplicity has an inverse. This framework allows us to explore deeper relationships and correlations among the multiplicities of the elements within X. By involving the finitely supported sets, we study the new defined group-valued multisets over infinite universes of discourse in a finitary manner. After presenting the algebraic groups in the framework of finitely supported sets, we study the finitely supported group-valued multisets. We provide a finitary characterization of group-valued multisets over infinite universes of discourse, and obtain new results that generalize the properties of hybrid sets obtained in the Zermelo\u2013Fraenkel framework.<\/jats:p>","DOI":"10.3390\/axioms14010031","type":"journal-article","created":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T11:24:19Z","timestamp":1735730659000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Group-Valued Multisets"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0563-8391","authenticated-orcid":false,"given":"Andrei","family":"Alexandru","sequence":"first","affiliation":[{"name":"Institute of Computer Science, Romanian Academy, 700505 Ia\u015fi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8166-9456","authenticated-orcid":false,"given":"Gabriel","family":"Ciobanu","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Romanian Academy, 700505 Ia\u015fi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/0001-8708(92)90011-9","article-title":"Sets with a negative number of elements","volume":"91","author":"Loeb","year":"1992","journal-title":"Adv. Math."},{"key":"ref_2","first-page":"1133","article-title":"Generalized multisets: From ZF to FSM","volume":"34","author":"Alexandru","year":"2015","journal-title":"Comput. Inform."},{"key":"ref_3","unstructured":"Barwise, J. (1977). Handbook of Mathematical Logic, North-Holland."},{"key":"ref_4","unstructured":"Jech, T. (1973). The Axiom of Choice, North-Holland. Studies in Logic and the Foundations of Mathematics."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1007\/BF01457986","article-title":"Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre","volume":"86","author":"Fraenkel","year":"1922","journal-title":"Math. Ann."},{"key":"ref_6","first-page":"27","article-title":"\u00dcber die Un\u00e4bhangigkeit des Auswahlsaxioms und Einiger seiner Folgerungen","volume":"31","author":"Lindenbaum","year":"1938","journal-title":"Comptes Rendus Des S\u00e9ances De La Soci\u00e9t\u00e9 Des Sci. Et Des Lett. De Vars."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Pitts, A. (2013). Nominal Sets Names and Symmetry in Computer Science, Cambridge University Press.","DOI":"10.1017\/CBO9781139084673"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bojanczyk, M., Klin, B., and Lasota, S. (2011, January 21\u201324). Automata with group actions. Proceedings of the 26th Symposium on Logic in Computer Science, Toronto, ON, Canada.","DOI":"10.1109\/LICS.2011.48"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Alexandru, A., and Ciobanu, G. (2020). Foundations of Finitely Supported Structures: A Set Theoretical Viewpoint, Springer.","DOI":"10.1007\/978-3-030-52962-8"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Alexandru, A., and Ciobanu, G. (2022). Soft sets with atoms. Mathematics, 10.","DOI":"10.3390\/math10121956"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Alexandru, A., and Ciobanu, G. (2016). Finitely Supported Mathematics: An Introduction, Springer.","DOI":"10.1007\/978-3-319-42282-4"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s00153-021-00787-2","article-title":"Various forms of infinity for finitely supported structures","volume":"61","author":"Alexandru","year":"2022","journal-title":"Arch. Math. Log."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T15:23:07Z","timestamp":1759850587000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,1]]},"references-count":12,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010031"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010031","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,1]]}}}