Skip to main content
SpringerLink
Log in

Alternative Proof of Standard Completeness Theorem for MTL

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

Abstract

As it was shown in Jenei and Montagna (Studia Logica 70(2): 183–192, 2002), Monoidal t-norm based logic (MTL) is a logic of left-continuous t-norms. In other words, this means that MTL enjoys the standard completeness theorem. In this paper we present a different proof of this theorem. In fact, we prove even more since we show that MTL is complete w.r.t. the class of standard MTL-algebras with finite congruence lattice or equivalently with finitely many Archimedean classes. We also show the connection between the congruence lattice of an MTL-chain and its Archimedean classes.

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

  1. Blok WJ, Alten CJ (2002) The finite embeddability property for residuated lattices, pocrims and BCK-algebras. Algebra Universalis 48:253–271

    Article  MathSciNet  MATH  Google Scholar 

  2. Blount K, Tsinakis C (2003) The structure of residuated lattices. Int J Algebra Comput 13(4):437–461

    Article  MathSciNet  MATH  Google Scholar 

  3. Esteva F, Godo G (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124(3):271–288

    Article  MathSciNet  MATH  Google Scholar 

  4. Esteva F, Gispert J, Godo G, Montagna F (2002) On the standard completeness of some axiomatic extensions of the monoidal t-norm logic. Studia Logica 71(2):199–226

    Article  MathSciNet  MATH  Google Scholar 

  5. Fuchs L (1963) Partially ordered algebraic systems. Pergamon Press, Oxford

    MATH  Google Scholar 

  6. Horčí k R (2005) Standard completeness theorem for Πmtl. Archive for Mathematical Logic 44(4):413–424

    Article  MathSciNet  Google Scholar 

  7. Horčí k R (2005) Algebraic properties of fuzzy logics. Ph. D. thesis, Czech Technical University in Prague

  8. Hrbacek K, Jech T (1999) Introduction to set theory. Monographs and textbooks in pure and applied mathematics, 3rd ed. Dekker, New York

    Google Scholar 

  9. Jenei S, Montagna F (2002) A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica 70(2):183–192

    Article  MathSciNet  MATH  Google Scholar 

  10. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Trends in logic, Vol 8. Kluwer, Dordrecht

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 2, 182 07, Prague, Czech Republic

    Rostislav Horčík

Authors
  1. Rostislav Horčík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rostislav Horčík.

Additional information

The author was supported by the Program “Information Society” under project 1ET100300517.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horčík, R. Alternative Proof of Standard Completeness Theorem for MTL. Soft Comput 11, 123–129 (2007). https://doi.org/10.1007/s00500-006-0058-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-006-0058-6

Keywords

Search

Navigation