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Associative Globally Monotone Extended Aggregation Functions

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Book cover Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 349))

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Abstract

In this contribution we deal with a global monotonicity condition for the class of associative extended aggregation functions. We insist on the idea that global monotonicity can be taken as a minimum requirement for an extended aggregation function to be considered consistent.

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Notes

  1. 1.

    Reflexive and transitive.

  2. 2.

    A t-norm is a two-variable function \(T:[0,1]^{2}\longrightarrow [0, 1]\) which is associative, symmetric, increasing in each variable and has neutral 1.

  3. 3.

    The minimum t-norm (\(T(x_{1},x_{2})=min(x_{1},x_{2}))\) is the only idempotent t-norm.

  4. 4.

    A t-conorm is a two-variable function \(S:[0, 1]^{2}\longrightarrow [0, 1]\) which is associative, symmetric, increasing in each variable and has neutral 0.

  5. 5.

    The maximum t-conorm (\(S(x_{1},x_{2})=max(x_{1},x_{2}))\) is the only idempotent t-conorm.

  6. 6.

    A mixed aggregation function exhibits different types of behavior on different parts of the domain.

  7. 7.

    A uninorm is a two-variable function \(U:[0,1]^{2}\longrightarrow [0, 1]\) which is associative, symmetric, increasing in each variable and has neutral e belonging to [0, 1].

  8. 8.

    A nullnorm is a two-variable function \(V:[0, 1]^{2}\longrightarrow [0, 1]\) which is associative, symmetric, increasing in each variable and has absorbent a belonging to [0, 1].

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Acknowledgments

The authors have written this contribution in tribute to Prof. Claudio Moraga in recognition of his important and extensive research in many areas of Soft Computing. This contribution has been partially supported by the Spanish Grant TIN2013-42795-P.

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Authors and Affiliations

  1. University of Alcalá, Alcalá de Henares, Spain

    Tomasa Calvo

  2. University of the Balearic Islands, Palma de Mallorca, Spain

    Gaspar Mayor & Jaume Suñer

Authors
  1. Tomasa Calvo
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  2. Gaspar Mayor
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  3. Jaume Suñer
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Corresponding author

Correspondence to Tomasa Calvo .

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Editors and Affiliations

  1. Escuela de IngenieríaInformática, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile

    Rudolf Seising

  2. Geschichte der Nat, "Ernst-Haeckel-Haus", Deutsches Museum, Munich, Germany

    Héctor Allende-Cid

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Calvo, T., Mayor, G., Suñer, J. (2017). Associative Globally Monotone Extended Aggregation Functions. In: Seising, R., Allende-Cid, H. (eds) Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing. Studies in Fuzziness and Soft Computing, vol 349. Springer, Cham. https://doi.org/10.1007/978-3-319-48317-7_18

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  • DOI: https://doi.org/10.1007/978-3-319-48317-7_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48316-0

  • Online ISBN: 978-3-319-48317-7

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