Skip to main content
SpringerLink
Log in

Free weak nilpotent minimum algebras

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

Abstract

We give a combinatorial description of the finitely generated free weak nilpotent minimum algebras and provide explicit constructions of normal forms.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  • Aguzzoli S, Bianchi M (2016a) On some questions concerning the axiomatisation of WNM-algebras and their subvarieties. Fuzzy Sets Systems 292:5–31

  • Aguzzoli S, Bianchi M (2016b) Single chain completeness and some related properties. Fuzzy Sets Systems 301:51–63

  • Aguzzoli S, Bova S, Gerla B (2012) Free Algebras and Functional Representation. In: Cintula P, Hajek P, Noguera C (eds) Handbook of Mathematical Fuzzy Logic. College Publications, London

    Google Scholar 

  • Aguzzoli S, Busaniche M, Marra V (2007) Spectral duality for finitely generated nilpotent minimum algebras, with applications. J Log Comput 17(4):749–765

    Article  MATH  MathSciNet  Google Scholar 

  • Aguzzoli S, Gerla B (2008) Normal forms and free algebras for some extensions of MTL. Fuzzy Sets Systems 159(10):1131–1152

    Article  MATH  MathSciNet  Google Scholar 

  • Bova S, Valota D (2012) Finite RDP-algebras: duality, coproducts and logic. J Log Comput 22:417–450

    Article  MATH  MathSciNet  Google Scholar 

  • Burris S, Sankappanvar H (1981) A Course in Universal Algebra. Springer-Verlag, New York

    Book  Google Scholar 

  • Busaniche M (2006) Free nilpotent minimum algebras. Math Log Quart 52:219–236

    Article  MATH  MathSciNet  Google Scholar 

  • Ciabattoni A, Galatos N, Terui K (2008) From axioms to analytic rules in nonclassical logics pp. 229–240 (2008). In: Proceedings of 23rd IEEE Symposium on Logic in Computer Science (LICS’08)

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

    Article  MATH  MathSciNet  Google Scholar 

  • Fodor J (1995) Nilpotent minimum and related connectives for fuzzy logic. In: Proceedings of FUZZ-IEEE’95, pp. 2077–2082

  • Gispert J (2003) Axiomatic extensions of the nilpotent minimum logic. Rep Math Log 37:113–123

    MATH  MathSciNet  Google Scholar 

  • Horn A (1969) Free L-algebras. J Symbol Log 34:475–480

    Article  MATH  Google Scholar 

  • Jenei S, Montagna F (2002) A proof of standard completeness for esteva and godo’s logic MTL. Stud Log 70(2):183–192

    Article  MATH  MathSciNet  Google Scholar 

  • Montagna F (2011) Completeness with respect to a chain and universal models in fuzzy logic. Arch Math Log 50:161–183

    Article  MATH  MathSciNet  Google Scholar 

  • Noguera C (2007) Algebraic study of axiomatic extensions oftriangular norm based fuzzy logics, vol 27. Monografies de l’Institutd’Investigació en Intel.ligència Artificial, Bellaterra

    Google Scholar 

  • Noguera C, Esteva F, Gispert J (2008) On triangular norm based axiomatic extensions of the weak nilpotent minimum logic. Math Log Quart 54(4):387–409

    Article  MATH  MathSciNet  Google Scholar 

  • Valota D (2010) Poset representation for free RDP-algebras. In: Hosni H, Montagna F (eds.) Probability, uncertainty and rationality, CRM Series, vol. 10. Edizioni della Scuola Normale Superiore (Pisa 2010)

  • Wang S (2007) A fuzzy logic for the revised drastic product t-norm. Soft comput 11(6):585–590

    Article  MATH  Google Scholar 

  • Wang S, Wang B, Pei D (2005) A fuzzy logic for anordinal sum t-norm. Fuzzy Sets Systems 149(2):297–307

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The second author is supported by the FWF Austrian Science Fund (Parameterized Compilation, P26200). The third author is supported by a Marie Curie INdAM-COFUND Outgoing Fellowship.

Author information

Authors and Affiliations

  1. Department of Computer Science, Università degli Studi di Milano, Via Comelico 39/41, 20135, Milano, Italy

    Stefano Aguzzoli

  2. Institut für Computergraphik und Algorithmen, Technische Universität, Favoritenstrasse 911, Wien, Austria

    Simone Bova

  3. Institut d’Investigació en Intel.ligència Artificial, Campus de la Universitat Autonoma de Barcelona, Bellaterra, Spain

    Diego Valota

Authors
  1. Stefano Aguzzoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simone Bova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Diego Valota
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Aguzzoli.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by A. Di Nola, D. Mundici, C. Toffalori, A. Ursini.

In memory of Franco Montagna.

Appendix: The 2-generated free WNM-algebra

Appendix: The 2-generated free WNM-algebra

In the following list, an item of the form,

$$\begin{aligned} 3>2: \mathbb {C}_{43}=0<y<y''xx''<y'x'<1\text {} \end{aligned}$$

represents a WNM-chain generated by x and y and specifies that \(\mathbb {C}_{43} \models x>y\), \(\mathrm {orbit}(\mathbb {C}_{43},x)=3\), and \(\mathrm {orbit}(\mathbb {C}_{43},y)=2\). In this format, the set,

$$\begin{aligned} \mathcal {K}_{2}=\{\mathbb {C}_i \mid i=1,2, \ldots , 74 \}\text {,} \end{aligned}$$

is as follows:

  • \(2<2\): \(\mathbb {C}_{1}=0<x<y<y''x''<y'x'<1\);

  • \(2<2\): \(\mathbb {C}_{2}=0<x<x''<y<y''<y'<x'<1\);

  • \(2<3\): \(\mathbb {C}_{3}=0<x<x''yy''<x'y'<1\);

  • \(2<3\): \(\mathbb {C}_{4}=0<x<x''<yy''<y'<x'<1\);

  • \(2<4\): \(\mathbb {C}_{5}=0<x<x''<y<y'y''<x'<1\);

  • \(2<5\): \(\mathbb {C}_{6}=0<x<x''<yy'y''<x'<1\);

  • \(2<6\): \(\mathbb {C}_{7}=0<x<x''y'<y<x'y''<1\);

  • \(2<6\): \(\mathbb {C}_{8}=0<y'<x<x''<x'<y<y''<1\);

  • \(2<6\): \(\mathbb {C}_{9}=0<x<x''<y'<y<y''<x'<1\);

  • \(2<7\): \(\mathbb {C}_{10}=0<x<x''y'<x'yy''<1\);

  • \(2<7\): \(\mathbb {C}_{11}=0<y'<x<x''<x'<yy''<1\);

  • \(2<7\): \(\mathbb {C}_{12}=0<x<x''<y'<yy''<x'<1\);

  • \(3<2\): \(\mathbb {C}_{13}=0<xx''<y<y''<y'<x'<1\);

  • \(3<3\): \(\mathbb {C}_{14}=0<xx''<yy''<y'<x'<1\);

  • \(3<4\): \(\mathbb {C}_{15}=0<xx''<y<y'y''<x'<1\);

  • \(3<5\): \(\mathbb {C}_{16}=0<xx''<yy'y''<x'<1\);

  • \(3<6\): \(\mathbb {C}_{17}=0<xx''y'<y<x'y''<1\);

  • \(3<6\): \(\mathbb {C}_{18}=0<y'<xx''<x'<y<y''<1\);

  • \(3<6\): \(\mathbb {C}_{19}=0<xx''<y'<y<y''<x'<1\);

  • \(3<7\): \(\mathbb {C}_{20}=0<xx''y'<x'yy''<1\);

  • \(3<7\): \(\mathbb {C}_{21}=0<y'<xx''<x'<yy''<1\);

  • \(3<7\): \(\mathbb {C}_{22}=0<xx''<y'<yy''<x'<1\);

  • \(4<4\): \(\mathbb {C}_{23}=0<x<y<x'x''y'y''<1\);

  • \(4<5\): \(\mathbb {C}_{24}=0<x<x'x''yy'y''<1\);

  • \(4<6\): \(\mathbb {C}_{25}=0<y'<x<x'x''<y<y''<1\);

  • \(4<7\): \(\mathbb {C}_{26}=0<y'<x<x'x''<yy''<1\);

  • \(5<6\): \(\mathbb {C}_{27}=0<y'<xx'x''<y<y''<1\);

  • \(5<7\): \(\mathbb {C}_{28}=0<y'<xx'x''<yy''<1\);

  • \(6<6\): \(\mathbb {C}_{29}=0<x'y'<x<y<x''y''<1\);

  • \(6<6\): \(\mathbb {C}_{30}=0<y'<x'<x<x''<y<y''<1\);

  • \(6<7\): \(\mathbb {C}_{31}=0<y'<x'<x<x''<yy''<1\);

  • \(6<7\): \(\mathbb {C}_{32}=0<x'y'<x<x''yy''<1\);

  • \(7<6\): \(\mathbb {C}_{33}=0<y'<x'<xx''<y<y''<1\);

  • \(7<7\): \(\mathbb {C}_{34}=0<y'<x'<xx''<yy''<1\);

  • \(2=2\): \(\mathbb {C}_{35}=0<xy<x''y''<x'y'<1\);

  • \(3=3\): \(\mathbb {C}_{36}=0<xx''yy''<x'y'<1\);

  • \(4=4\): \(\mathbb {C}_{37}=0<xy<x'x''y'y''<1\);

  • \(5=5\): \(\mathbb {C}_{38}=0<xx'x''yy'y''<1\);

  • \(6=6\): \(\mathbb {C}_{39}=0<x'y'<xy<x''y''<1\);

  • \(7=7\): \(\mathbb {C}_{40}=0<x'y'<xx''yy''<1\);

  • \(2>2\): \(\mathbb {C}_{41}=0<y<x<x''y''<x'y'<1\);

  • \(2>2\): \(\mathbb {C}_{42}=0<y<y''<x<x''<x'<y'<1\);

  • \(3>2\): \(\mathbb {C}_{43}=0<y<y''xx''<y'x'<1\);

  • \(3>2\): \(\mathbb {C}_{44}=0<y<y''<xx''<x'<y'<1\);

  • \(4>2\): \(\mathbb {C}_{45}=0<y<y''<x<x'x''<y'<1\);

  • \(5>2\): \(\mathbb {C}_{46}=0<y<y''<xx'x''<y'<1\);

  • \(6>2\): \(\mathbb {C}_{47}=0<y<y''x'<x<y'x''<1\);

  • \(6>2\): \(\mathbb {C}_{48}=0<x'<y<y''<y'<x<x''<1\);

  • \(6>2\): \(\mathbb {C}_{49}=0<y<y''<x'<x<x''<y'<1\);

  • \(7>2\): \(\mathbb {C}_{50}=0<y<y''x'<y'xx''<1\);

  • \(7>2\): \(\mathbb {C}_{51}=0<x'<y<y''<y'<xx''<1\);

  • \(7>2\): \(\mathbb {C}_{52}=0<y<y''<x'<xx''<y'<1\);

  • \(2>3\): \(\mathbb {C}_{53}=0<yy''<x<x''<x'<y'<1\);

  • \(3>3\): \(\mathbb {C}_{54}=0<yy''<xx''<x'<y'<1\);

  • \(4>3\): \(\mathbb {C}_{55}=0<yy''<x<x'x''<y'<1\);

  • \(5>3\): \(\mathbb {C}_{56}=0<yy''<xx'x''<y'<1\);

  • \(6>3\): \(\mathbb {C}_{57}=0<yy''x'<x<y'x''<1\);

  • \(6>3\): \(\mathbb {C}_{58}=0<x'<yy''<y'<x<x''<1\);

  • \(6>3\): \(\mathbb {C}_{59}=0<yy''<x'<x<x''<y'<1\);

  • \(7>3\): \(\mathbb {C}_{60}=0<yy''x'<y'xx''<1\);

  • \(7>3\): \(\mathbb {C}_{61}=0<x'<yy''<y'<xx''<1\);

  • \(7>3\): \(\mathbb {C}_{62}=0<yy''<x'<xx''<y'<1\);

  • \(4>4\): \(\mathbb {C}_{63}=0<y<x<y'y''x'x''<1\);

  • \(5>4\): \(\mathbb {C}_{64}=0<y<y'y''xx'x''<1\);

  • \(6>4\): \(\mathbb {C}_{65}=0<x'<y<y'y''<x<x''<1\);

  • \(7>4\): \(\mathbb {C}_{66}=0<x'<y<y'y''<xx''<1\);

  • \(6>5\): \(\mathbb {C}_{67}=0<x'<yy'y''<x<x''<1\);

  • \(7>5\): \(\mathbb {C}_{68}=0<x'<yy'y''<xx''<1\);

  • \(6>6\): \(\mathbb {C}_{69}=0<y'x'<y<x<y''x''<1\);

  • \(6>6\): \(\mathbb {C}_{70}=0<x'<y'<y<y''<x<x''<1\);

  • \(7>6\): \(\mathbb {C}_{71}=0<x'<y'<y<y''<xx''<1\);

  • \(7>6\): \(\mathbb {C}_{72}=0<y'x'<y<y''xx''<1\);

  • \(6>7\): \(\mathbb {C}_{73}=0<x'<y'<yy''<x<x''<1\);

  • \(7>7\): \(\mathbb {C}_{74}=0<x'<y'<yy''<xx''<1\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aguzzoli, S., Bova, S. & Valota, D. Free weak nilpotent minimum algebras. Soft Comput 21, 79–95 (2017). https://doi.org/10.1007/s00500-016-2340-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-016-2340-6

Keywords

Search

Navigation