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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:
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