Skip to main content
SpringerLink
Log in

The stable topology for residuated lattices

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

Abstract

The spectrum of a residuated lattice L is the set Spec(L) of all prime i-filters. It is well known that Spec(L) can be endowed with the spectral topology. The main scope of this paper is to introduce and study another topology on Spec(L), the so called stable topology, which turns out to be coarser than the spectral one. With this and in view, we introduce the notions of pure i-filter for a residuated lattice and the notion of normal residuated lattice. So, we generalize to case of residuated lattice some results relative to MV-algebras (Belluce and Sessa in Quaest Math 23:269–277, 2000; Cavaccini et al. in Math Japonica 45(2):303–310, 1997) or BL-algebras (Eslami and Haghani in Kybernetika 45:491–506, 2009; Leustean in Central Eur J Math 1(3): 382–397, 2003; Turunen and Sessa in Mult-Valued Log 6(1–2):229–249, 2001).

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

  • Balbes R, Dwinger Ph (1974) Distributive lattices, XIII. University of Missouri Press, Columbia

  • Belluce LP (1986) Semisimple algebras of infinite-valued logic and Bold fuzzy set theory. Can J Math 38(6):1356–1379

    Article  MathSciNet  MATH  Google Scholar 

  • Belluce LP, Sessa S (2000) The stable topology for MV-algebras. Quaest Math 23:269–277

    Article  MathSciNet  MATH  Google Scholar 

  • Buşneag D, Piciu D (2006) Residuated lattices of fractions relative to an ∧-closed system. Bull Math Soc Math Roumanie Tome 49, (97)(1):13–24

    Google Scholar 

  • Buşneag D, Piciu D, Jeflea A (2010) Archimedean residuated lattices. Ann St Univ A I Cuza Iasi Matematica (SN) Tome LVI f 2:227–252

    Google Scholar 

  • Cavaccini C, Cella C, Georgescu G (1997) Pure ideals of MV-algebras. Math Japonica 45(2):303–310

    MathSciNet  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, Dordrecht

  • Ciungu LC (2006) Classes of residuated lattices. An Univ Craiova Ser Mat Inf 33:180–207

    MathSciNet  MATH  Google Scholar 

  • Cornish WH (1972) Normal lattices. J Austr Math Soc 14:200–215

    Article  MathSciNet  MATH  Google Scholar 

  • Costinescu O (1969) Elemente de topologie generală. Ed. Tehnică, Bucureşti

  • De Marco G (1983) Projectivity of pure ideals. Rend Sem Math Univ Padova 68:289–304

    MathSciNet  Google Scholar 

  • El-Ezeh H (1990) Topological characterization of certain classes of lattices. Rend Sem Mat Univ Padova 83:13–18

    MathSciNet  Google Scholar 

  • Eslami E, Haghani FKh (2009) Pure filters and stable topology on BL-algebras. Kybernetika 45:491–506

    MathSciNet  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 151. Elsevier

  • Georgescu G, Leustean L (2002) Some classes of pseudo-BL algebras. J Austral Math Soc 73:1–27

    Article  MathSciNet  Google Scholar 

  • Georgescu G, Voiculescu I (1987) Isomorphic sheaf representations of normal lattices. J Pure Appl Algebra 45:213–223

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Iorgulescu A (2009) Algebras of logic as BCK algebras. ASE, Bucharest

  • Johnstone PT (1982) Stone spaces. Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge

  • Leustean L (2003) The prime and maximal spectra and the reticulation of BL-algebras. Central Eur J Math 1(3):382–397

    Article  MathSciNet  MATH  Google Scholar 

  • Muresan C (2008) The reticulation of a residuated lattice. Bull Math Soc Sci Math Roumanie 51 (99) (1):47–65

    Google Scholar 

  • Piciu D (2007) Algebras of fuzzy logic. Ed. Universitaria, Craiova

  • Turunen E (1999) Mathematics behind fuzzy logic. Physica-Verlag

  • Turunen E, Sessa S (2001) Local BL algebras. Mult-Valued Log 6(1–2):229–249

    MathSciNet  MATH  Google Scholar 

  • Wallman H (1938) Lattices and topological spaces. Ann Math 39(2):112–126

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge the anonymous reviewer for helpful comments and suggestions. Following the suggestion of the referee, the authors have in intention to extend the results of this paper for the non-commutative case.

Author information

Authors and Affiliations

  1. Faculty of Exacte Sciences, Department of Mathematics, University of Craiova, 13, Al. I. Cuza st., 200585, Craiova, Romania

    Cătălin Buşneag & Dana Piciu

Authors
  1. Cătălin Buşneag
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dana Piciu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cătălin Buşneag.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buşneag, C., Piciu, D. The stable topology for residuated lattices. Soft Comput 16, 1639–1655 (2012). https://doi.org/10.1007/s00500-012-0849-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-012-0849-x

Keywords

Search

Navigation