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State operators on GMV algebras

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Abstract

Flaminio and Montagna recently introduced state \(MV\) algebras as \(MV\) algebras with an internal state in the form of a unary operation. Di Nola and Dvurečenskij further presented a stronger variation of state \(MV\) algebras called state-morphism \(MV\) algebras. In the paper we present state \(GMV\) algebras and state-morphism \(GMV\) algebras which are non-commutative generalizations of the mentioned algebras.

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References

  • Chang CC (1958) Algebraic analysis of many valued logic. Trans Am Math Soc 88:467–490

    MATH  Google Scholar 

  • Di Nola A, Dvurečenskij (2009) A State-morphism \(MV\) algebras. Ann Pure Appl Logic 161:161–173

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

    Article  MATH  Google Scholar 

  • Dvurečenskij A (2002) Pseudo MV algebras are intervals in \(\ell\)-groups. J Aust Math Soc 72:427–445

    Article  MATH  MathSciNet  Google Scholar 

  • Flaminio T, Montagna F (2007) An algebraic approach to states on \(MV\) algebras. In: Štěpnička M, Novák V, Bodenhofer U (eds) Proceedings of EUSFLAT 2007, vol 2, pp 201–206

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

    Article  MATH  MathSciNet  Google Scholar 

  • Flondor P, Leustean I (2004) \(MV\) algebras with operators (the commutative and the non-commutative case). Discrete Math 274:41–76

    Article  MATH  MathSciNet  Google Scholar 

  • Georgescu G, Iorgulescu A (2001) Pseudo MV algebras. Multiple Valued Logic 6:95–135

    MATH  MathSciNet  Google Scholar 

  • Goodearl KR (1986) Partially ordered abelian groups with interpolation. Mathematical surveys and monographs no. 20, American Mathematical Society, Providence

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

    MATH  MathSciNet  Google Scholar 

  • Leustean I (2006) Non-commutative Łukasiewicz propositional logic. Arch Math Logic 45:191–213

    Article  MATH  MathSciNet  Google Scholar 

  • Mundici D (1986) Interpretation of AF \(C^*\) -algebras in sentential calculus. J Funct Analys 65:15–63

    Article  MATH  MathSciNet  Google Scholar 

  • Mundici D (1995) Averaging the truth-value in Łukasiewicz logic. Stud Logic 55:113–127

    Article  MATH  MathSciNet  Google Scholar 

  • Rachůnek J (2002a) A non-commutative generalization of MV algebras. Czech Math J 52:255–273

    Article  MATH  Google Scholar 

  • Rachůnek J (2002b) Prime spectra of non-commutative generalizations of \(MV\) algebras. Algebra Univ 48:151–169

    Article  MATH  Google Scholar 

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Acknowledgments

The authors are very indebted to the anonymous referees for their valuable comments and suggestions which helped to improve the paper. The first author was supported by the Council of Czech Government, MSM 6198959214.

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Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova 40, 779 00, Olomouc, Czech Republic

    Jiří Rachůnek

  2. Department of Mathematical Methods in Economy, Faculty of Economics, VŠB-Technical University Ostrava, Sokolská 33, 701 21, Ostrava, Czech Republic

    Dana Šalounová

Authors
  1. Jiří Rachůnek
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  2. Dana Šalounová
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Correspondence to Jiří Rachůnek.

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Rachůnek, J., Šalounová, D. State operators on GMV algebras. Soft Comput 15, 327–334 (2011). https://doi.org/10.1007/s00500-010-0568-0

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  • DOI: https://doi.org/10.1007/s00500-010-0568-0

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