# Article

 Title: Łukasiewicz tribes are absolutely sequentially closed bold algebras (English) Author: Frič, Roman Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 52 Issue: 4 Year: 2002 Pages: 861-874 Summary lang: English . Category: math . Summary: We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean $MV$-algebra) can be uniquely extended to a sequentially continuous measure on the generated Łukasiewicz tribe and, in a natural way, the extension is maximal. We prove that for normed measures on Łukasiewicz tribes monotone (sequential) continuity implies sequential continuity, hence the assumption of sequential continuity is not restrictive. This yields a characterization of the Łukasiewicz tribes as bold algebras absolutely sequentially closed with respect to the extension of probabilities. The result generalizes the relationship between fields of sets and the generated $\sigma$-fields discovered by J. Novák. We introduce the category of bold algebras and sequentially continuous homomorphisms and prove that Łukasiewicz tribes form an epireflective subcategory. The restriction to fields of sets yields the epireflective subcategory of $\sigma$-fields of sets. (English) Keyword: $MV$-algebra Keyword: bold algebra Keyword: field of sets Keyword: Łukasiewicz tribe Keyword: sequential convergence Keyword: sequential continuity Keyword: measure Keyword: extension of measures Keyword: sequential envelope Keyword: absolute sequentially closed bold algebra Keyword: epireflective subcategory MSC: 06B35 MSC: 18B99 MSC: 28E10 MSC: 28E15 MSC: 54C20 MSC: 60A10 idZBL: Zbl 1016.28013 idMR: MR1940065 . Date available: 2009-09-24T10:57:33Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/127770 . 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