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Decision Making under Constraints pp 71–82Cite as

Generalized Ordinal Sum Constructions of t-norms on Bounded Lattices

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Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 276))

Abstract

In this contribution, we propose an ordinal sum construction of t-norms on bounded lattices from a system of lattices (endowed by t-norms) whose index set forms a bounded lattice. The ordinal sum construction is determined by a lattice-based sum of bounded lattices and interior operators.

This research was partially supported from the ERDF/ESF project AI-Met4AI (No. CZ.02.1.01/0.0/0.0/17_049/0008414). The additional support was also provided by the Czech Science Foundation through the project of No.18-06915S.

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Notes

  1. 1.

    This is a consequence of the fact that \(0_L\) and \(1_L\) are nullary operations in the bounded lattice L.

  2. 2.

    Note that if an index set P is a linearly ordered set of indices with the bottom and top elements, then P is clearly a bounded lattice.

  3. 3.

    Here we consider the definition of lattice as a partially ordered set such that any two elements have an infimum and a supremum. Note that the algebraic definition of lattice and the definition based on a partially ordered set are equivalent (see, e.g., [12, 20]).

  4. 4.

    The closure operator is defined dually to the interior operator, i.e., \(h:L\rightarrow L\) is a closure operator if 1. \(h(0_L)=0_L\), 2. \(h(h(x))=h(x)\), 3. \(h(x\vee y)=h(x)\vee h(y)\), 4. \(x\le h(x)\) holds for any \(x,y\in L\). For details, we refer to [6].

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Author information

Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, CE IT4Innovations, 30. dubna 22, 701 03, Ostrava, Czech Republic

    Antonín Dvořák & Michal Holčapek

Authors
  1. Antonín Dvořák
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  2. Michal Holčapek
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Corresponding author

Correspondence to Michal Holčapek .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

    Martine Ceberio

  2. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

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Dvořák, A., Holčapek, M. (2020). Generalized Ordinal Sum Constructions of t-norms on Bounded Lattices. In: Ceberio, M., Kreinovich, V. (eds) Decision Making under Constraints. Studies in Systems, Decision and Control, vol 276. Springer, Cham. https://doi.org/10.1007/978-3-030-40814-5_9

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