{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T13:30:18Z","timestamp":1777469418193,"version":"3.51.4"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2015,3,1]],"date-time":"2015-03-01T00:00:00Z","timestamp":1425168000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/ http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18], respectively. In the second section, definitions of a ring and a \u03c3-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a \u03c3-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between \u03c3-ring and \u03c3-algebra of sets are formalized and some related examples are given. The formalization is based on [15], and also referred to [9] and [16].<\/jats:p>","DOI":"10.2478\/forma-2015-0004","type":"journal-article","created":{"date-parts":[[2015,7,14]],"date-time":"2015-07-14T07:49:39Z","timestamp":1436860179000},"page":"51-57","source":"Crossref","is-referenced-by-count":9,"title":["\u03c3-ring and \u03c3-algebra of Sets\n                    <sup>1<\/sup>"],"prefix":"10.2478","volume":"23","author":[{"given":"Noboru","family":"Endou","sequence":"first","affiliation":[{"name":"Gifu National College of Technology, Gifu, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kazuhisa","family":"Nakasho","sequence":"additional","affiliation":[{"name":"Shinshu University, Nagano, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Shinshu University, Nagano, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2015,3,31]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/23\/1\/article-p51.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2015-0004","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:36:29Z","timestamp":1777383389000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2015-0004"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,3,31]]},"published-print":{"date-parts":[[2015,3,1]]}},"alternative-id":["10.2478\/forma-2015-0004"],"URL":"https:\/\/doi.org\/10.2478\/forma-2015-0004","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3,1]]}}}