{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:52:56Z","timestamp":1777517576977,"version":"3.51.4"},"reference-count":40,"publisher":"SAGE Publications","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["COM"],"published-print":{"date-parts":[[2024,2,14]]},"abstract":"<jats:p>We study the two algebras of complemented subsets that were introduced in the constructive development of the Daniell approach to measure and integration within Bishop-style constructive mathematics. We present their main properties both for the so-called here categorical complemented subsets and for the extensional complemented subsets. We translate constructively the classical bijection between subsets and Boolean-valued, total functions by establishing a bijection between complemented subsets (categorical or extensional) and Boolean-valued, partial functions (categorical or extensional). The role of Myhill\u2019s axiom of non-choice in the equivalence between categorical and extensional subsets is discussed. We introduce swap algebras of type (I) and (II) as an abstract version of Bishop\u2019s algebras of complemented subsets of type (I) and (II), respectively, and swap rings as an abstract version of the structure of Boolean-valued partial functions on a set. Our examples of swap algebras and swap rings together with the included here results indicate that their theory is a certain generalisation of the theory of Boolean algebras and Boolean rings, a fact which we find interesting both from a constructive and a classical point of view.<\/jats:p>","DOI":"10.3233\/com-230462","type":"journal-article","created":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T10:29:35Z","timestamp":1708079375000},"page":"1-33","source":"Crossref","is-referenced-by-count":1,"title":["Complemented subsets and Boolean-valued, partial functions"],"prefix":"10.1177","author":[{"given":"Daniel","family":"Misselbeck-Wessel","sequence":"first","affiliation":[{"name":"Alfried Krupp Wissenschaftskolleg Greifswald, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Iosif","family":"Petrakis","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Verona, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","reference":[{"key":"10.3233\/COM-230462_ref1","doi-asserted-by":"crossref","unstructured":"M.J.\u00a0Beeson, Formalizing constructive mathematics: Why and how, in: Constructive Mathematics, LNM, Vol.\u00a0873, Springer-Verlag, 1981, pp.\u00a0146\u2013190.","DOI":"10.1007\/BFb0090733"},{"key":"10.3233\/COM-230462_ref2","doi-asserted-by":"crossref","unstructured":"M.J.\u00a0Beeson, Foundations of Constructive Mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, 1985.","DOI":"10.1007\/978-3-642-68952-9_2"},{"key":"10.3233\/COM-230462_ref3","unstructured":"E.\u00a0Bishop, Foundations of Constructive Analysis, McGraw-Hill, 1967."},{"key":"10.3233\/COM-230462_ref4","doi-asserted-by":"crossref","unstructured":"E.\u00a0Bishop and D.S.\u00a0Bridges, Constructive Analysis, Grundlehren der math. 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