Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Classes of examples of pseudo-MV algebras, pseudo-BL algebras and divisible bounded non-commutative residuated lattices

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

Abstract

We illustrate by classes of examples the close connections existing between pseudo-MV algebras, on the one hand, and pseudo-BL algebras and divisible bounded non-commutative residuated lattices, on the other hand. We use equivalent definitions of these algebras, as particular cases of pseudo-BCK algebras. We analyse the strongness, the pseudo-involutive center and the filters for each example.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Buşneag D, Iorgulescu A, Rudeanu S (2008) On filters and deductive systems in algebra of logic (submitted)

  • Ceterchi R (1999) On algebras with implications, categorically equivalent to pseudo-MV algebras. In: The proceedings of the fourth international symposium on economic informatics. INFOREC Printing House, Bucharest, Romania, pp 912–916

  • Ceterchi R (2001) Pseudo-Wajsberg Algebras (A special issue dedicated to the memory of Gr. C. Moisil). Mult Val Logic 6(1–2):67–88

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Di Nola A, Georgescu G, Iorgulescu A (2002a) Pseudo-BL algebras: part I. Multi Val Logic 8(5–6):673–714

    MATH  MathSciNet  Google Scholar 

  • Di Nola A, Georgescu G, Iorgulescu A (2002b) Pseudo-BL algebras: part II. Multi Val Logic 8(5–6):717–750

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Dvurecenskij A, Giuntini R, Kowalski T (2009) On the structure of pseudo BL-algebras and pseudo hoops in quantum logics (submitted)

  • Dymek G (2006) Bipartite pseudo MV-algebras. Discuss Math Gen Algebra Appl 26:183–197

    MATH  MathSciNet  Google Scholar 

  • Dymek G, Walendziak A (2007) On maximal ideals of pseudo MV-algebras. Comment Math 47:117–126

    MATH  MathSciNet  Google Scholar 

  • Flondor P, Georgescu G, Iorgulescu A (2001) Pseudo-t-norms and pseudo-BL algebras. Soft Comput 5(5):355–371

    Article  MATH  Google Scholar 

  • Font JM, Rodriguez AJ, Torrens A (1984) Wajsberg algebras. Stochastica VIII(1):5–31

    MathSciNet  Google Scholar 

  • Georgescu G, Iorgulescu A (1999) Pseudo-MV Algebras: a noncommutative extension of MV algebras. In: The proceedings of the fourth international symposium on economic informatics. Bucharest, Romania, pp 961–968

  • Georgescu G, Iorgulescu A (2000) Pseudo-BL algebras: a noncommutative extension of BL algebras. In: Abstracts of the fifth international conference FSTA 2000. Slovakia, pp 90–92

  • Georgescu G, Iorgulescu A (2001a) Pseudo-BCK algebras: an extension of BCK algebras. In: Proceedings of DMTCS’01: combinatorics, computability and logic. Springer, London, pp 97–114

  • Georgescu G, Iorgulescu A (2001b) Pseudo-MV algebras. Mult Val Log 6(1–2):95–135

    MATH  MathSciNet  Google Scholar 

  • Hájek P (1996) Metamathematics of fuzzy logic. Institute of Computer Science, Academy of Science of Czech Report, Technical report 682

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

    MATH  Google Scholar 

  • Iorgulescu A (2004a) On pseudo-BCK algebras and porims. Sci Math Japonicae 10(16):293–305

    MathSciNet  Google Scholar 

  • Iorgulescu A (2004b) Classes of BCK algebras: Part III, Preprint Series of the Institute of Mathematics of the Romanian Academy, preprint nr. 3/2004, pp 1–37

  • Iorgulescu A (2005) Pseudo-Iséki algebras. Connection with pseudo-BL algebras. J Multiple Valued Log Soft Comput 11(3–4):263–308

    MATH  MathSciNet  Google Scholar 

  • Iorgulescu A (2006a) Classes of pseudo-BCK algebras: part I. J Multiple Valued Log Soft Comput 12(1–2):71–130

    MATH  MathSciNet  Google Scholar 

  • Iorgulescu A (2006b) Classes of pseudo-BCK algebras: part II. J Multiple Valued Log Soft Comput 12(5–6):575–629

    MATH  MathSciNet  Google Scholar 

  • Iorgulescu A (2008a) On BCK algebras—part II: new algebras. The ordinal sum (product) of two bounded BCK algebras. Soft Comput 12(9):835–856. doi:10.1007/s00500-007-0239-y (Online first)

    Article  MATH  Google Scholar 

  • Iorgulescu A (2008b) On BCK algebras—part III: classes of examples of proper MV algebras, BL algebras and divisible bounded residuated lattices, with or without condition (WNM) (submitted)

  • Iséki K (1966) An algebra related with a propositional calculus. Proc Jpn Acad 42:26–29

    Article  MATH  Google Scholar 

  • Mundici D (1986a) MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math Jpn 31(6):889–894

    MATH  MathSciNet  Google Scholar 

  • Mundici D (1986b) Interpretation of AF C-algebras in Lukasiewicz sentential calculus. J Funct Anal 65:15–63

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Swamy KLN (1965) Dually residuated lattice ordered semigroups. Math Ann 159:105–114

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

All my gratitude to the anonimous reviewer for his kind, valuable and useful suggestions, which all helped me to improve technically the paper and also its presentation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Afrodita Iorgulescu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iorgulescu, A. Classes of examples of pseudo-MV algebras, pseudo-BL algebras and divisible bounded non-commutative residuated lattices. Soft Comput 14, 313–327 (2010). https://doi.org/10.1007/s00500-009-0405-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-009-0405-5

Keywords

Navigation