×

Łukasiewicz tribes are absolutely sequentially closed bold algebras. (English) Zbl 1016.28013

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.

MSC:

28E10 Fuzzy measure theory
60A10 Probabilistic measure theory
06B35 Continuous lattices and posets, applications
18B99 Special categories
54C20 Extension of maps
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] L. P. Belluce: Semisimple algebras of infinite valued logics. Canad. J. Math. 38 (1986), 1356-1379. · Zbl 0625.03009
[2] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publ., Dordrecht, 2000. · Zbl 0937.06009
[3] A. Dvurečenskij: Loomis-Sikorski theorem for \(\sigma \)-complete \(MV\)-algebras and \(l\)-groups. Austral. Math. Soc. J. Ser. A 68 (2000), 261-277. · Zbl 0958.06006
[4] R. Frič: A Stone-type duality and its applications to probability. Proceedings of the 12th Summer Conference on General Topology and its Applications, North Bay, August 1997, 1999, pp. 125-137. · Zbl 0945.54012
[5] R. Frič: Boolean algebras: convergence and measure. Topology Appl. 111 (2001), 139-149. · Zbl 0977.54004
[6] H. Herrlich and G. E. Strecker: Category theory. Second edition. Heldermann Verlag, Berlin, 1976.
[7] J. Jakubík: Sequential convergences on \(MV\)-algebras. Czechoslovak Math. J. 45(120) (1995), 709-726. · Zbl 0845.06009
[8] M. Jurečková: The measure extension theorem on \(MV\)-algebras. Tatra Mt. Math. Publ. 6 (1995), 55-61. · Zbl 0859.28009
[9] E. P. Klement: Characterization of finite fuzzy measures using Markoff-kernels. J. Math. Anal. Appl. 75 (1980), 330-339. · Zbl 0471.60008
[10] E. P. Klement and M. Navara: A characterization of tribes with respect to the Łukasiewicz \(t\)-norm. Czechoslovak Math. J. 47(122) (1997), 689-700. · Zbl 0902.28015
[11] F. Kôpka and F. Chovanec: \(D\)-posets. Math. Slovaca 44 (1994), 21-34. · Zbl 0789.03048
[12] P. Kratochvíl: Multisequences and measure. General Topology and its Relations to Modern Analysis and Algebra, IV (Proc. Fourth Prague Topological Sympos., 1976), Part B Contributed Papers, Society of Czechoslovak Mathematicians and Physicists, Praha, 1977, pp. 237-244.
[13] R. Mesiar: Fuzzy sets and probability theory. Tatra Mt. Math. Publ. 1 (1992), 105-123. · Zbl 0790.60005
[14] J. Novák: Über die eindeutigen stetigen Erweiterungen stetiger Funktionen. Czechoslovak Math. J. 8(83) (1958), 344-355. · Zbl 0087.37501
[15] J. Novák: On the sequential envelope. General Topology and its Relation to Modern Analysis and Algebra (I) (Proc. (First) Prague Topological Sympos., 1961), Publishing House of the Czechoslovak Academy of Sciences, Praha, 1962, pp. 292-294. · Zbl 0139.16001
[16] J. Novák: On convergence spaces and their seqeuntial envelopes. Czechoslovak Math. J. 15(90) (1965), 74-100.
[17] J. Novák: On sequential envelopes defined by means of certain classes of functions. Czechoslovak Math. J. 18(93) (1968), 450-456. · Zbl 0164.23401
[18] E. Pap: Null-Additive Set Functions. Kluwer Acad. Publ., Dordrecht, Ister Science, Bratislava, 1995. · Zbl 0968.28010
[19] K. Piasecki: Probability fuzzy events defined as denumerable additive measure. Fuzzy Sets and Systems 17 (1985), 271-284. · Zbl 0604.60005
[20] P. Pták and S. Pulmannová: Orthomodular Structures as Quantum Logics. Kluwer Acad. Publ., Dordrecht, 1991. · Zbl 0743.03039
[21] B. Riečan and T. Neubrunn: Integral, Measure, and Ordering. Kluwer Acad. Publ., Dordrecht-Boston-London, 1997. · Zbl 0916.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.