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Generalized state operators on residuated lattices

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Abstract

We give a definition of a generalized state operator (g-state operator) \(\sigma \) on a residuated lattice X and a g-state residuated lattice \((X,\sigma )\), by which the class of all g-state residuated lattices is proved to be a variety, and consider properties of g-state residuated lattices. We prove some fundamental results about them, such as characterizations of \(\sigma \)-filters, extended \(\sigma \)-filters, homomorphism theorems for g-state residuated lattices. Moreover, we show that every g-state residuated lattice is a subdirect product of subdirectly irreducible g-state residuated lattices.

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References

  • Ciungu LC (2008) Bosbach and Riečan states on residuated lattices. J Appl Funct Anal 3:175–188

    MathSciNet  MATH  Google Scholar 

  • Constantinescu NM (2014) State filters on fuzzy structures with internal states. Soft Comput 18:1841–1852

    Article  MATH  Google Scholar 

  • Constantinescu NM (2012) On pseudo-BL algebras with internal state. Soft Comput 16:1915–1922

    Article  MATH  Google Scholar 

  • Dvurečenskij A (2001) States on pseudo MV-algebras. Studia Log 68:301–327

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A, Rachůnek J (2006) Probabilistic averaging in bounded R\({\ell }\)-monoids. Semigroup Forum 72:190–206

    MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A, Rachůnek J (2006) On Riečan and Bosbach states for bounded non-commutative R\({\ell }\)-monoids. Math Slovaca 56:487–500

    MathSciNet  MATH  Google Scholar 

  • Dvurěcenskij A, Rachůnek J, Šalounova D (2012) State operators on generalizations of fuzzy structures. Fuzzy Sets Syst 187:58–76

    Article  MathSciNet  MATH  Google Scholar 

  • Flaminio T, Montanga F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50:138–152

    Article  MathSciNet  MATH  Google Scholar 

  • Georgescu G (2004) Bosbach states on fuzzy structures. Soft Comput 8:217–230

    Article  MATH  Google Scholar 

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics, studies in logic and the foundations of mathematics. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Haveshki M, Mohamadhasani M (2012) Extended filters in bounded commutative R\(\ell \)-monoids. Soft Comput 16:2165–2173

    Article  MATH  Google Scholar 

  • Hart JB, Rafter J, Tsinakis C (2002) The structure of commutative residuated lattices. Int J Algebra Comput 12:509–524

    Article  MathSciNet  MATH  Google Scholar 

  • Haveshki M, Saeid AB, Eslami E (2006) Some types of filters in BL algebras. Soft Comput 10:657–664

    Article  MATH  Google Scholar 

  • He P, Xin X, Yang Y (2015) On state residuated lattices. Soft Comput 19:2083–2094

    Article  MATH  Google Scholar 

  • Kondo M (2014) Characterization of extended filters in residuated lattices. Soft Comput 18:427–432

    Article  MATH  Google Scholar 

  • Kondo M (2014) States on bounded commutative residuated lattices. Math Slovaca 64:1093–1104

    Article  MathSciNet  MATH  Google Scholar 

  • Kondo M, Kawaguchi MF (2016) Some properties of generalized state operators on residuated lattices. Proc IEEE ISMVL 2016:162–166

    MathSciNet  Google Scholar 

  • Kondo M, Watari O, Kawaguchi MF, Miyakoshi M (2007) A logic determined by commutative residuated lattices. EUSFLAT Conference, pp 45–48

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

    MathSciNet  MATH  Google Scholar 

  • Rachůnek J, Salounova D (2011) State operators on GMV-algebras. Soft Comput 15:327–334

    Article  MATH  Google Scholar 

  • Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40:467–473

    Article  MathSciNet  MATH  Google Scholar 

  • Ward M, Dilworth RP (1939) Residuated lattices. Trans AMS 45:335–354

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This study was funded by JSPS KAKENHI (Grant Number 15K00024).

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Authors and Affiliations

  1. Tokyo Denki University, Tokyo, Japan

    Michiro Kondo

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  1. Michiro Kondo
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Correspondence to Michiro Kondo.

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The author declares that he has no conflict of interest.

Additional information

Communicated by A. Di Nola.

This work was partly supported by JSPS KAKENHI Grant Number 15K00024.

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Kondo, M. Generalized state operators on residuated lattices. Soft Comput 21, 6063–6071 (2017). https://doi.org/10.1007/s00500-016-2324-6

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  • DOI: https://doi.org/10.1007/s00500-016-2324-6

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