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F-transforms and semimodule homomorphisms

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Abstract

Relationships between F-transforms based on variants of fuzzy partitions and \({\mathcal {R}}\)-semimodules homomorphisms are investigated, where \({\mathcal {R}}\) are appropriate semirings defined on complete residuated lattices or MV-algebras. For general form of a fuzzy partition, these two notions are proved to be identical.

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References

  • Bede B, Rudas IJ (2011) Approximation properties of fuzzy transforms. Fuzzy Sets Syst 180(1):20–40

    Article  MathSciNet  MATH  Google Scholar 

  • Berstel J, Perrin D (1985) Theory of codes. Academic Press, Orlando, Florida

    MATH  Google Scholar 

  • Chang CC (1958) Algebraic analysis of many-valued logic. Trans. A.M.S. 93:74–80

    Google Scholar 

  • Di Martino F et al (2007) Compression and decompression of images with discrete fuzzy transforms. Inf Sci 177(11):2349–2362

    Article  MathSciNet  MATH  Google Scholar 

  • Di Martino F et al (2008) An image coding/decoding method based on direct and inverse fuzzy tranforms. Int J Approx Reason 48:110–131

    Article  MATH  Google Scholar 

  • Di Martino F et al (2010) A segmentation method for images compressed by fuzzy transforms. Fuzzy Sets Syst 161(1):56–74

    Article  MathSciNet  MATH  Google Scholar 

  • Di Martino F et al (2010) Fuzzy transforms method and attribute dependency in data analysis. Inf Sci 180(4):493–505

    Article  MathSciNet  MATH  Google Scholar 

  • Di Martino F et al (2011) Fuzzy transforms method in prediction data analysis. Fuzzy Sets Syst 180(1):146–163

    Article  MathSciNet  MATH  Google Scholar 

  • Di Nola A, Gerla B (2005) Algebras of Łukasiewicz logic and their semiring reducts. Contemp Math 377:131–144

    Article  MATH  Google Scholar 

  • Di Nola A, Lettieri A, Perfilieva I, Novák V (2007) Algebraic analysis of fuzzy systems. Fuzzy Sets Syst 158:1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Golan JS (1999) Semirings and their applications. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Golan JS (2013) Power algebras over semirings: with applications in mathematics and computer science. Springer, Dordrecht

    Google Scholar 

  • Khastan A, Perfilieva I, Alijani Z (2016) A new fuzzy approximation method to Cauchy problem by fuzzy transform. Fuzzy Sets Syst 288:75–95

    Article  MathSciNet  MATH  Google Scholar 

  • Močkoř J (2017) Spaces with fuzzy partitions and fuzzy transform. Soft Comput 13(21):3479–3492

    MATH  Google Scholar 

  • Močkoř J (2018) Powerset operators of extensional fuzzy sets. Iran J Fuzzy Syst 15(2):143–163

    MathSciNet  MATH  Google Scholar 

  • Močkoř J (2018) Axiomatic of lattice-valued F-transform. Fuzzy Sets Syst 342:53–66

    Article  MathSciNet  MATH  Google Scholar 

  • Močkoř J, Holčapek M (2017) Fuzzy objects in spaces with fuzzy partitions. Soft Comput 21(24):7269–7284

    Article  MATH  Google Scholar 

  • Močkoř J, Hurtik P (2018) Lattice-valued F-transform and similarity relations. Fuzzy Sets Syst 342:67–89

    Article  MathSciNet  MATH  Google Scholar 

  • Novák V, Perfilijeva I, Močkoř J (1991) Mathematical principles of fuzzy logic. Kluwer, Boston

    Google Scholar 

  • Perfilieva I (2006) Fuzzy transforms and their applications to image compression. Lecture Notes in Computer Science 19–31

  • Perfilieva I (2006) Fuzzy transforms: theory and applications. Fuzzy Sets Syst 157:993–1023

    Article  MathSciNet  MATH  Google Scholar 

  • Perfilieva I, Novak V, Dvořak A (2008) Fuzzy transforms in the analysis of data. Int J Approx Reason 48:36–46

    Article  MATH  Google Scholar 

  • Perfilieva I, Singh AP, Tiwari SP (2015) On the relationship among \(F\)-transform, fuzzy rough set and fuzzy topology. In: Proceedings of IFSA-EUSFLAT. Amsterdam, Atlantis Press, pp 1324–1330

  • Rodabaugh SE (1997) Powerset operator based foundation for point-set lattice theoretic (poslat) fuzzy set theories and topologies. Quaestiones Mathematicae 20(3):463–530

    Article  MathSciNet  MATH  Google Scholar 

  • Stefanini L (2011) F-transform with parametric generalized fuzzy partitions. Fuzzy Sets Syst 180(1):98–120

    Article  MathSciNet  MATH  Google Scholar 

  • Štěpnička M, Valašek R (2005) Numerical solution of partial differential equations with the help of fuzzy transform. In: Proceedings of the FUZZ-IEEE, Reno, Necada, pp 1104–1009

  • Tan Y-J (2014) Bases in semimodules over commutative semirings. Linear Algebra Appl 443:139–152

    Article  MathSciNet  MATH  Google Scholar 

  • Tomasiello S (2017) An alternative use of fuzzy transform with application to a class of delay differential equations. Int J Comput Math 94(9):1719–1726

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This study was partially supported by the Project GA18-06915S provided by the Grant Agency of the Czech Republic.

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  1. Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Jiří Močkoř

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  1. Jiří Močkoř
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Correspondence to Jiří Močkoř.

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Communicated by A. Di Nola.

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Močkoř, J. F-transforms and semimodule homomorphisms. Soft Comput 23, 7603–7619 (2019). https://doi.org/10.1007/s00500-019-03766-1

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