| 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:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127770 |
| . |
| Reference:
|
[1] L. P. Belluce: Semisimple algebras of infinite valued logics.Canad. J. Math. 38 (1986), 1356–1379. MR 0873417, 10.4153/CJM-1986-069-0 |
| Reference:
|
[2] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning.Kluwer Academic Publ., Dordrecht, 2000. MR 1786097 |
| Reference:
|
[3] A. Dvurečenskij: Loomis-Sikorski theorem for $\sigma $-complete $MV$-algebras and $l$-groups.Austral. Math. Soc. J. Ser. A 68 (2000), 261–277. MR 1738040, 10.1017/S1446788700001993 |
| Reference:
|
[4] R. Frič: A Stone-type duality and its applications to probability.In: Proceedings of the 12th Summer Conference on General Topology and its Applications, North Bay, August 1997, 1999, pp. 125–137. MR 1718934 |
| Reference:
|
[5] R. Frič: Boolean algebras: convergence and measure.Topology Appl. 111 (2001), 139–149. MR 1806034, 10.1016/S0166-8641(99)00195-9 |
| Reference:
|
[6] H. Herrlich and G. E. Strecker: Category theory. Second edition.Heldermann Verlag, Berlin, 1976. MR 2377903 |
| Reference:
|
[7] J. Jakubík: Sequential convergences on $MV$-algebras.Czechoslovak Math. J. 45(120) (1995), 709–726. MR 1354928 |
| Reference:
|
[8] M. Jurečková: The measure extension theorem on $MV$-algebras.Tatra Mt. Math. Publ. 6 (1995), 55–61. MR 1363983 |
| Reference:
|
[9] E. P. Klement: Characterization of finite fuzzy measures using Markoff-kernels.J. Math. Anal. Appl. 75 (1980), 330–339. Zbl 0471.60008, MR 0581822, 10.1016/0022-247X(80)90083-9 |
| Reference:
|
[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. MR 1479313, 10.1023/A:1022822719086 |
| Reference:
|
[11] F. Kôpka and F. Chovanec: $D$-posets.Math. Slovaca 44 (1994), 21–34. MR 1290269 |
| Reference:
|
[12] P. Kratochvíl: Multisequences and measure.In: 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. MR 0460576 |
| Reference:
|
[13] R. Mesiar: Fuzzy sets and probability theory.Tatra Mt. Math. Publ. 1 (1992), 105–123. Zbl 0790.60005, MR 1230469 |
| Reference:
|
[14] J. Novák: Über die eindeutigen stetigen Erweiterungen stetiger Funktionen.Czechoslovak Math. J. 8(83) (1958), 344–355. MR 0100826 |
| Reference:
|
[15] J. Novák: On the sequential envelope.In: 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. MR 0175082 |
| Reference:
|
[16] J. Novák: On convergence spaces and their seqeuntial envelopes.Czechoslovak Math. J. 15(90) (1965), 74–100. MR 0175083 |
| Reference:
|
[17] J. Novák: On sequential envelopes defined by means of certain classes of functions.Czechoslovak Math. J. 18(93) (1968), 450–456. MR 0232335 |
| Reference:
|
[18] E. Pap: Null-Additive Set Functions.Kluwer Acad. Publ., Dordrecht, Ister Science, Bratislava, 1995. Zbl 0968.28010, MR 1368630 |
| Reference:
|
[19] K. Piasecki: Probability fuzzy events defined as denumerable additive measure.Fuzzy Sets and Systems 17 (1985), 271–284. MR 0819364, 10.1016/0165-0114(85)90093-4 |
| Reference:
|
[20] P. Pták and S. Pulmannová: Orthomodular Structures as Quantum Logics.Kluwer Acad. Publ., Dordrecht, 1991. MR 1176314 |
| Reference:
|
[21] B. Riečan and T. Neubrunn: Integral, Measure, and Ordering.Kluwer Acad. Publ., Dordrecht-Boston-London, 1997. MR 1489521 |
| . |
