Skip to main content
SpringerLink
Log in

Extension of domains of states

  • Original Paper
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

We deal with categorical aspects of the extensions of generalized probability measures. In particular, we study various domains of fuzzy sets, describe the relationships between σ-fields of crisp sets and generated Łukasiewicz tribes of measurable functions, and mention some probabilistic aspects. D-posets and sequential continuity play an important role.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Adámek J (1983). Theory of mathematical structures. Reidel, Dordrecht

    MATH  Google Scholar 

  • Bugajski S (2001). Statistical maps I. Basic properties. Math Slovaca 51: 321–342

    MATH  MathSciNet  Google Scholar 

  • Bugajski S (2001). Statistical maps II. Operational random variables. Math Slovaca 51: 343–361

    MATH  MathSciNet  Google Scholar 

  • Butnariu D and Klement EP (1991). Triangular norm-based measures and their Markov-kernel representation. J Math Anal Appl 162: 111–143

    Article  MATH  MathSciNet  Google Scholar 

  • di Nola A and Navara M (2005). The σ-complete MV-algebras which have enough states. Colloq Math 103: 121–130

    Article  MATH  MathSciNet  Google Scholar 

  • Dvurečenskij A and Pulmannová S (2000). New trends in quantum structures. Kluwer/Ister Science, Dordrecht/Bratislava

    MATH  Google Scholar 

  • Foulis DJ and Bennett MK (1994). Effect algebras and unsharp quantum logics. Found Phys 24: 1331–1352

    Article  MathSciNet  Google Scholar 

  • Frič R (1988). Remarks on sequential envelopes. Rend Istit Math Univ Trieste 26: 604–612

    Google Scholar 

  • Frič R (2001). Boolean algebras: convergence and measure. Topol Appl 111: 139–149

    MATH  Google Scholar 

  • Frič R (2002a). Convergence and duality. Appl Categorical Struct 10: 257–266

    Article  MATH  Google Scholar 

  • Frič R (2002b). Łukasiewicz tribes are absolutely sequentially closed bold algebras. Czechoslovak Math J 52: 861–874

    Article  MATH  MathSciNet  Google Scholar 

  • Frič R (2002c). Measures on MV-algebras. Soft Comput 7: 130–137

    Article  MATH  Google Scholar 

  • Frič R (2004a). Duality for generalized events. Math Slovaca 54: 49–60

    MATH  MathSciNet  Google Scholar 

  • Frič R (2004b). Coproducts of D-posets and their applications to probability. Int J Theoret Phys 43: 1625–1632

    Article  MATH  Google Scholar 

  • Frič R (2005a). Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mountains Math Publ 30: 21–34

    MATH  Google Scholar 

  • Frič R (2005b). Extension of measures: a categorical approach. Math Bohemica 130: 397–407

    MATH  Google Scholar 

  • Frič R (2007). Statistical maps: a categorical approach. Math Slovaca 57: 41–57

    Article  MATH  MathSciNet  Google Scholar 

  • Frič R, Chovanec F (2006) On D-posets of fuzzy sets (Abstracts of the 20th Summer Conference on Real Functions Theory, Liptovský Ján, September 10–15, 2006, MÚ SAV, Košice, 2006)

  • Frič R and Hušek M (1983). Projectively generated convergence of sequences. Czechoslovak Math J 33: 525–536

    MathSciNet  Google Scholar 

  • Frič R and Jakubík J (2001). Sequential convergences on Boolean algebras defined by systems of maximal filters. Czechoslovak Math J 51: 261–274

    Article  MATH  MathSciNet  Google Scholar 

  • Gudder S (1998). Fuzzy probability theory. Demonstratio Math 31: 235–254

    MATH  MathSciNet  Google Scholar 

  • Jurečková M (1995). The measure extension theorem for MV-algebras. Tatra Mountains Math Publ 6: 56–61

    Google Scholar 

  • Koutník V (1967). On sequentially regular convergence spaces. Czechoslovak Math J 17: 232–247

    MathSciNet  Google Scholar 

  • Kôpka F (1992). D-posets of fuzzy sets. Tatra Mountains Math Publ 1: 83–87

    MATH  Google Scholar 

  • Kôpka F and Chovanec F (1994). D-posets. Math Slovaca 44: 21–34

    MATH  MathSciNet  Google Scholar 

  • Kratochvíl P (1976) Multisequences and measure. In: General Topology and its Relations to Modern Analysis and Algeba IV (Proc. Fourth Prague Topological Sympos., 1976), Part B Contributed Papers, Society of Czechoslovak Mathematicians and Physicists, Praha, pp 237–244

  • Mesiar R (1992). Fuzzy sets and probability theory. Tatra Mountains Math Publ 1: 105–123

    MATH  MathSciNet  Google Scholar 

  • Novák J (1958). Ueber die eindeutigen stetigen Erweiterungen stetiger Funktionen. Czechoslovak Math J 8: 344–355

    MathSciNet  Google Scholar 

  • Novák J (1962) On the sequential envelope. In: General Topology and its Relations to Modern Analysis and Algeba (I) (Proc. (First) Prague Topological Sympos., 1961), Publishing House of the Czechoslovak Academy of Sciences, Prague, 1962, pp 292–294

  • Novák J (1965). On convergence spaces and their sequential envelopes. Czechoslovak Math J 15: 74–100

    MathSciNet  Google Scholar 

  • Novák J (1968). On sequential envelopes defined by means of certain classes of functions. Czechoslovak Math J 18: 450–456

    MathSciNet  Google Scholar 

  • Papčo M (2004). On measurable spaces and measurable maps. Tatra Mountains Math Publ 28: 125–140

    MATH  Google Scholar 

  • Papčo M (2005). On fuzzy random variables: examples and generalizations. Tatra Mountains Math Publ 30: 175–185

    MATH  Google Scholar 

  • Papčo M (2007). On effect algebras. Soft Comput 12: 373–379

    Article  Google Scholar 

  • Pták P and Pulmannová S (1991). Orthomodular structures as quantum logics. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Riečan B and Mundici D (2002). Probability on MV-algebras. In: Pap E, (eds) Handbook of Measure Theory, vol II, pp 869–910. North-Holland, Amsterdam

    Google Scholar 

  • Riečan B and Neubrunn T (1997). Integral, measure and ordering. Kluwer, Dordrecht

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovak Republic

    Roman Frič

  2. Catholic University in Ružomberok, Nám. A. Hlinku 56/1, 034 01, Ružomberok, Slovak Republic

    Roman Frič

Authors
  1. Roman Frič
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman Frič.

Additional information

This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0071-06, Protocol for Scientific and Technological Collaboration between the Republic of Italy and Slovak Republic, Project No. 15: Algebraic and Logical Systems of Soft Computing, and VEGA 2/6088/26.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frič, R. Extension of domains of states. Soft Comput 13, 63–70 (2009). https://doi.org/10.1007/s00500-008-0293-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-008-0293-0

Keywords

Search

Navigation