Skip to main content
Springer Nature Link
Log in

Analysis of the \(\varphi \)-Index of Inclusion Restricted to a Set of Indexes

  • Conference paper
  • First Online:
Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2024)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 1174))

  • 87 Accesses

Abstract

The \(\varphi \)-index of inclusion has proven to be a suitable generalization of the inclusion in the fuzzy setting. In this paper, the properties of the \(\varphi \)-index of inclusion, when its definition is restricted to a subset of indexes, are analyzed. The theoretical results obtained in this work are necessary in order to develop fuzzy inference systems based on the \(\varphi \)-index of inclusion.

Partially supported by the Spanish Science Ministry project PGC2018-095869-B-I00, co-funded by the European Regional Development Fund (ERDF). Partially supported by the 2014–2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in project PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124.

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

Access this chapter

Log in via an institution

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

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Usually, in approaches related to measures of inclusion, e.g., [10], bijective mappings in \(\mathcal {U}\) are called transformations.

  2. 2.

    A pair of functions (f, g) forms an isotone Galois Connection if the following holds: \(f(x)\le y\) if and only of \(x\le g(y)\), for all \(x,y\in [0,1]\).

References

  1. Bustince, H., Madrid, N., Ojeda-Aciego, M.: The notion of weak-contradiction: definition and measures. IEEE Trans. Fuzzy Syst. 23, 1057–1069 (2015)

    Article  Google Scholar 

  2. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  3. Fan, J., Xie, W., Pei, J.: Subsethood measure: new definitions. Fuzzy Sets Syst. 106, 201–209 (1999)

    Article  MathSciNet  Google Scholar 

  4. Kitainik, L.M.: Fuzzy inclusions and fuzzy dichotomous decision procedures. In: Kacprzyk, J., Orlovski, S.A. (eds.) Optimization Models Using Fuzzy Sets and Possibility Theory, vol. 4, pp. 154–170. Springer, Dordrecht (1987). https://doi.org/10.1007/978-94-009-3869-4_11

    Chapter  Google Scholar 

  5. Madrid, N., Ojeda-Aciego, M.: A View of f-indexes of inclusion under different axiomatic definitions of fuzzy inclusion. In: Moral, S., Pivert, O., Sánchez, D., Marín, N. (eds.) SUM 2017. LNCS (LNAI), vol. 10564, pp. 307–318. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67582-4_22

    Chapter  Google Scholar 

  6. Madrid, N., Ojeda-Aciego, M.: Functional degrees of inclusion and similarity between L-fuzzy sets. Fuzzy Sets Syst. 390, 1–22 (2020)

    Article  MathSciNet  Google Scholar 

  7. Madrid, N., Ojeda-Aciego, M.: The f-index of inclusion as optimal adjoint pair for fuzzy modus ponens. Fuzzy Sets Syst. 466, 108474 (2023)

    Article  MathSciNet  Google Scholar 

  8. Madrid, N., Ojeda-Aciego, M.: Approaching the square of opposition in terms of the f-indexes of inclusion and contradiction. Fuzzy Sets Syst. 476, 108769 (2024)

    Article  MathSciNet  Google Scholar 

  9. Madrid, N., Ojeda-Aciego, M., Perfiliev, I.: F-inclusion indexes between fuzzy sets. In: Proceedings of IFSA-EUSFLAT (2015)

    Google Scholar 

  10. Sinha, D., Dougherty, E.: Fuzzification of set inclusion: theory and applications. Fuzzy Sets Syst. 55, 15–42 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Depto. de Matemáticas, Universidad de Cádiz, Cádiz, Spain

    Nicolás Madrid & Eloísa Ramírez-Poussa

Authors
  1. Nicolás Madrid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eloísa Ramírez-Poussa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolás Madrid .

Editor information

Editors and Affiliations

  1. LIP6, BC169, Sorbonne Université, Paris, France

    Marie-Jeanne Lesot

  2. Departamento de Engenharia Mecânica, University of Lisbon, Lisbon, Portugal

    Susana Vieira

  3. Electrical and Computer Engineering, University of Alberta, Edmonton, Canada

    Marek Z. Reformat

  4. INESC-ID,Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

    João Paulo Carvalho

  5. INESC-ID, Av. das Forças Armadas, 1649-026, ISCTE - University Institute of Lisbon, Lisboa, Portugal

    Fernando Batista

  6. LIP6, BC169, Place Jussieu 4, 75005, Sorbonne University, Paris, France

    Bernadette Bouchon-Meunier

  7. Machine Intelligence Institute, 715 North Avenue, Iona Collage, NEW ROCHELLE, NY, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Madrid, N., Ramírez-Poussa, E. (2024). Analysis of the \(\varphi \)-Index of Inclusion Restricted to a Set of Indexes. In: Lesot, MJ., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2024. Lecture Notes in Networks and Systems, vol 1174. Springer, Cham. https://doi.org/10.1007/978-3-031-74003-9_1

Download citation

Publish with us

Policies and ethics