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On the residuation principle of n-dimensional R-implications

  • Fuzzy systems and their mathematics
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Abstract

The essential idea of the residuation principle plays a fundamental role in the residuated lattice theory based on R-implications, which are derived from left-continuous t-norms providing a general logical framework frequently applied to achieve solutions for multiple criteria and decision-making problems. The definition of n-dimensional fuzzy R-implications (n-DRI) is introduced, showing that the main properties of R-implications on the unit interval [0,1] can be preserved in the n-dimensional upper simplex \(L_n([0,1])\), based on left-continuous n-dimensional t-norms. We also show a construction method for this class of implications, studying its relationships with intrinsic properties such as identity, neutrality, ordering and exchange principles. In addition, by considering the action of n-dimensional automorphisms, the conjugate of n-dimensional fuzzy R-implications is studied. The characterization of n-dimensional fuzzy R-implications and a methodology to obtain these operators from n-dimensional aggregations on \(L_n([0,1])\) is discussed, as the left-continuous n-dimensional t-norms. The representable Łukasiewicz implication and minimum aggregation on \(L_n([0,1])\) are considered to compare multiple alternatives in both approaches: (i) using admissible linear orders in \(L_n([0,1])\) provided by a sequence of aggregations and (ii) applying the arithmetic means in n-dimensional data application, based on multiple attributes related to a selection of the best CIM (computer-integrated manufacturing) software systems obtained from decision maker evaluations. The theoretical results on n-DRI are carries out to the fuzzy module evaluation in cloud computing environments.

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Notes

  1. The operator \(+:L_n([0,1])^2 \rightarrow L_n([0,1])\) is given as \({\mathbf {x}} +{\mathbf {y}} = (\min (x_1+y_1,1), \ldots , \min (x_n+y_n,1))\), \(\forall {\mathbf {x}}=(x_1, \ldots , x_n), {\mathbf {y}}=(y_1, \dots , y_n) \in L_n([0,1])\).

  2. https://github.com/brunomourapaz/CloudSim.

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Acknowledgements

This work was partially supported by Brazilian Funding Agency CAPES, CNPq (311429/2020-3 and 309160/2019-7) also including PqG/FAPERGS (21/2551-0002057-1).

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Authors and Affiliations

  1. Center of Technological Development, Federal University of Pelotas, Pelotas, RS, Brazil

    Rosana Zanotelli, Bruno Moura & Renata Reiser

  2. Department of Informatics and Applied Mathematics, Federal University of Rio Grande do Norte, Natal, RN, Brazil

    Benjamin Bedregal

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  1. Rosana Zanotelli
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  2. Bruno Moura
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  3. Renata Reiser
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Correspondence to Renata Reiser.

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Zanotelli, R., Moura, B., Reiser, R. et al. On the residuation principle of n-dimensional R-implications. Soft Comput 26, 8403–8426 (2022). https://doi.org/10.1007/s00500-022-07221-6

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