Skip to main content
SpringerLink
Log in

Algebraic study to generalized Bosbach states on residuated lattices

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Generalized Bosbach states of type I and II, which are also called type I and II states, are useful for the development of algebraic theory of probabilistic models for fuzzy logics. In this paper, a pure algebraic study to the generalization of Bosbach states on residuated lattices is made. By rewriting the equations of Bosbach states, an alternative definition of type II states is given, and five types of generalized Bosbach states of type III, IV, V, VI and VII (or simply, type III, IV, V, VI and VII states) are introduced. The relationships among these generalized Bosbach states and properties of them are investigated by some examples and results. Particularly, type IV states are a new type of generalized Bosbach states which are different from type I, II and III states; type V (resp. VI) states can be equivalently defined by both type I (resp. II) states and type IV states; type I, II and III states are equivalent when the codomain is an MV-algebra as well as type V and type VI states.

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

  • Bahls P, Cole J, Galatos N, Jipsen P, Tsinakis C (2003) Cancellative residuated lattices. Algebra Universalis 50:83–106

    Article  MathSciNet  MATH  Google Scholar 

  • Borzooei RA, Dvurec̆enskij A, Zahiri O (2014) State BCK-algebras and state-morphism BCK-algebras. Fuzzy Sets Syst 244:86–105

    Article  MathSciNet  Google Scholar 

  • Bĕlohlávek R, Vychodil V (2005) Fuzzy Equ Log. Springer, New York

    Google Scholar 

  • Botur M, Halas̆ R, Kühr J (2012) States on commutative basic algebras. Fuzzy Sets Syst 187:77–91

    Article  MATH  Google Scholar 

  • Botur M, Dvurec̆enskij A (2013) State-morphism algebras–general approach. Fuzzy Sets Syst 218:90–102

    Article  MathSciNet  MATH  Google Scholar 

  • Busneag C (2010) States on Hilbert algebras. Stud Log 94:177–188

    Article  MathSciNet  MATH  Google Scholar 

  • Ciungu LC (2008) Bosbach and Riĕcan states on residuated lattices. J Appl Funct Anal 2:175–188

    MathSciNet  Google Scholar 

  • Ciungu LC (2008) States on pseudo-BCK algebras. Math. Rep. 10(60):17–36

    MathSciNet  Google Scholar 

  • Ciungu LC, Dvurec̆enskij A, Hycko M (2011) State BL-algebras. Soft Comput 15:619–634

    Article  MATH  Google Scholar 

  • Ciungu LC, Georgescu G, Mureşan C (2013) Generalized Bosbach states: part I. Arch Math Log 52:335–376

    Article  MATH  Google Scholar 

  • Ciungu LC, Georgescu G, Mureşan C (2013) Generalized Bosbach states: part II. Arch Math Log 52:707–732

    Article  MATH  Google Scholar 

  • Ciungu LC (2014) Non-commutative multiple-valued logic algebras. Springer, New York

    Book  MATH  Google Scholar 

  • Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning, trends in logic-studia logica library 7. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  • Di Nola A, Dvurec̆enskij A, Lettieri A (2010) On varieties of MV-algebras with internal states. Int J Approx Reason 51:680–694

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurec̆enskij A, Rachunek J (2006) Probabilistic averaging in bounded R\(l\)-monoids. Semigroup Forum 72:191–206

    Article  MathSciNet  Google Scholar 

  • Dvurec̆enskij A, Rachunek J (2006) On Riec̆an and Bosbach states for bounded non-commutative R\(l\)-monoids. Mathematica Slovaca 56:487–500

    MathSciNet  Google Scholar 

  • Dvurec̆censkij A, Rachunek J (2006) Bounded commutative residuated R\(l\)-monoids with general comparability and states. Soft Comput 10:212–218

    Article  Google Scholar 

  • Dvurec̆enskij A (2001) States on Pseudo-MV-algebras. Stud Log 68:301–329

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Dvurec̆enskij A, Rachunek J, S̆alounová D (2012) Erratum to State operators on generalizations of fuzzy structures [Fuzzy Sets and Systems 187 (2012) 58–76]. Fuzzy Sets Syst 194:97–99

    Article  Google Scholar 

  • Dvurec̆enskij A (2007) Every linear pseudo BL-algebra admits a state. Soft Comput 11:495–501

    Article  Google Scholar 

  • Dvurec̆enskij A (2011) States on quantum structures versus integrals. Int J Theor Phys 50:3761–3777

    Article  MathSciNet  Google Scholar 

  • Esteva F, Godo L (2001) Monoidal \(t\)-norm based logic: towards a logic for left-continuous \(t\)-norms. Fuzzy Sets Syst 124:271–288

    Article  MathSciNet  MATH  Google Scholar 

  • Flaminio T, Montagna F (2007) An algebraic approach to states on MV-algebras. In: Novák V (ed) Fuzzy Logic 2. Proceedings of the 5th EUSFLAT Conference, Ostrava, vol. II, Sept 11–14, pp 201–206

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

    Article  MathSciNet  MATH  Google Scholar 

  • Georgescu G, Mureşan C (2010) Generalized Bosbach states. Available at http://arxiv.org/abs/1007.2575

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

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  • Jenc̆ová A, Pulmannová S (2015) Effect algebras with state operator. Fuzzy Sets Syst 260:43–61

  • Kroupa T (2006) Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst 157:2771–2782

    Article  MathSciNet  MATH  Google Scholar 

  • Kroupa T (2006) Representation and extension of states on MV-algebras. Arch Math Log 45:381–392

    Article  MathSciNet  MATH  Google Scholar 

  • Liu L, Zhang X (2008) States on R\(_0\)-algebras. Soft Comput 12:1099–1104

    Article  MATH  Google Scholar 

  • Liu L (2011) States on finite monoidal \(t\)-norm based algebras. Inf Sci 181:1369–1383

    Article  MATH  Google Scholar 

  • Ma ZM, Hu BQ (2014) Characterizations and new subclasses of \({\cal {I}}\)-filters in residuated lattices. Fuzzy Sets Syst 247:92–107

    Article  MathSciNet  Google Scholar 

  • Ma ZM (2014) Two types of MTL-L-filters in residuated lattices. J Intel Fuzzy Syst 27:681–689

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Mertanen J, Turunen E (2008) States on semi-divisible generalized residuated lattices reduce to states on MV-algebras. Fuzzy Sets Syst 159:3051–3064

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Riecan B (2000) On the probability on BL-algebras. Acta Math Nitra 4:3–13

    Google Scholar 

  • Turunen E, Mertanen J (2008) States on semi-divisible residuated lattices. Soft Comput 12:353–357

    Article  MATH  Google Scholar 

  • Zhou H, Zhao B (2012) Generalized Bosbach and Riec̆an states based on relative negations in residuated lattices. Fuzzy Sets Syst 187:33–57

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao B, Zhou H (2013) Generalized Bosbach and Riec̆an states on nucleus-based-Glivenko residuated lattices. Arch Math Log 52:689–706

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous reviewers for their valuable suggestions in improving this paper. This research was supported by Macao Science and Technology Development Fund MSAR. (Ref. 018/2014/A1), the NSF of Shandong Province (Grant No. ZR2013FL006) and the AMEP (DYSP) of Linyi University.

Author information

Authors and Affiliations

  1. School of Science, Linyi University, Linyi, 276005, People’s Republic of China

    Zhen Ming Ma & Zun Wei Fu

Authors
  1. Zhen Ming Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zun Wei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zun Wei Fu.

Additional information

Communicated by A. Di Nola.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, Z.M., Fu, Z.W. Algebraic study to generalized Bosbach states on residuated lattices. Soft Comput 19, 2541–2550 (2015). https://doi.org/10.1007/s00500-015-1671-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-015-1671-z

Keywords

Search

Navigation