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Some Remarks on Idempotent Nullnorms on Bounded Lattices

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Aggregation Functions in Theory and in Practice (AGOP 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 581))

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Abstract

Nullnorms are generalizations of triangular norms (t-norms) and triangular conorms (t-conorms) with a zero element to be an arbitrary point from an arbitrary bounded lattice. In this paper, we study on the existence of idempotent nullnorms on bounded lattices. We show that there exists unique idempotent nullnorm on an arbitrary distributive bounded lattice. We prove that an idempotent nullnorm may not always exist on every bounded lattice. Furthermore, we propose the construction method to obtain idempotent nullnorms on a bounded lattice under additional assumptions on given zero element. As by-product of this method, we see that it is in existence an idempotent nullnorm on non-distributive bounded lattices.

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Acknowledgment

The full proofs of the theorems in this paper are contained in the paper [6]. We are grateful to the anonymous reviewers and editors for their valuable comments which have enabled us to improve the original version of our paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey

    Gül Deniz Çaylı & Funda Karaçal

Authors
  1. Gül Deniz Çaylı
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  2. Funda Karaçal
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Corresponding author

Correspondence to Gül Deniz Çaylı .

Editor information

Editors and Affiliations

  1. School of Informatics, University of Skövde, Skövde, Sweden

    Vicenç Torra

  2. Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar

  3. KERMIT Research Unit Knowledge-based Systems, Ghent University, Ghent, Belgium

    Bernard De Baets

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Çaylı, G.D., Karaçal, F. (2018). Some Remarks on Idempotent Nullnorms on Bounded Lattices. In: Torra, V., Mesiar, R., Baets, B. (eds) Aggregation Functions in Theory and in Practice. AGOP 2017. Advances in Intelligent Systems and Computing, vol 581. Springer, Cham. https://doi.org/10.1007/978-3-319-59306-7_4

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  • DOI: https://doi.org/10.1007/978-3-319-59306-7_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59305-0

  • Online ISBN: 978-3-319-59306-7

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