Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices | Studia Logica
Skip to main content
Springer Nature Link
Log in

Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

In any variety of bounded integral residuated lattice-ordered commutative monoids (bounded residuated lattices for short) the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties (WL k) k>0 defined in Cignoli and Torrens (Studia Logica 100:1107–1136, 2012 [7]).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /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 includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bergman, C., Universal Algebra. Fundamental and Selected topics, Pure and Applied Mathematics, CRC Press, Chapman & Hall, 2012.

  2. Blok, W. J., and I. M. A. Ferreirim, On the structure of hoops, Algebra Universalis 43:233–157, 2000.

  3. Blok, W. J., and D. Pigozzi, Algebraizable logics, vol.77, n.396, Mem. Amer. Math. Soc., Providence, 1989.

  4. Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, Springer, New York, 1981.

  5. Castaño D., Díaz-Varela J. P., Torrens A.: Indecomposability of free algebras in some subvarieties of residuated lattices and their bounded subreducts. Soft Computing 15, 1449–1455 (2011)

    Article  Google Scholar 

  6. Cignoli R., Torrens A.: Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic. Archive for Mathematical Logic 42, 361–370 (2003)

    Article  Google Scholar 

  7. Cignoli R., Torrens A.: Varieties of commutative integral bounded residuated lattices admitting a Boolean retraction term. Studia Logica 100, 1107–1136 (2012)

    Article  Google Scholar 

  8. Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, New York (2007)

    Google Scholar 

  9. Georgescu G., Leuştean L., Mureşan C.: Maximal residuated lattices with lifting boolean center. Algebra Universalis 63(1), 83–99 (2010)

    Article  Google Scholar 

  10. Gispert J., Torrens A.: Bounded BCK-algebras and their generated variety. Mathematical Logic Quarterly 53, 41–59 (2007)

    Article  Google Scholar 

  11. Höhle, U., Commutative, residuated l-monoids, in U. Höhle and E. P. Klement (eds.), Non-classical Logics and their Applications to Fuzzy Subsets: A Handbook on the Mathematical Foundations of Fuzzy Set Theory, Kluwer, Boston, 1995, pp. 53–106.

  12. Kowalski T.: Semisimplicity, EDPC and discriminator varieties of residuated lattices. Studia Logica 77(2), 255–265 (2004)

    Article  Google Scholar 

  13. Kowalski T., Ono H.: The variety of residuated lattices is generated by its finite simple members. Reports on Mathematical Logic 34, 57–75 (2000)

    Google Scholar 

  14. Kowalski, T., and H. Ono, Residuated lattices: An algebraic glimpse at logics without contraction, Preliminary Report, 2001.

  15. McCune, W., Prover9-Mace4, http://www.cs.unm.edu/~mccune/prover9/.

  16. Torrens A.: Semisimplicity and the discriminator in bounded BCK-algebras. Algebra Universalis 63, 1–16 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Antoni Torrens

Authors
  1. Antoni Torrens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antoni Torrens.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Torrens, A. Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices. Stud Logica 104, 849–867 (2016). https://doi.org/10.1007/s11225-016-9655-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-016-9655-2

Keywords

Search

Navigation