On the residuation principle of n-dimensional R-implications

• Published in: Soft computing (Berlin, Germany) Vol. 26; no. 17; pp. 8403 - 8426
• Main Authors: Zanotelli, Rosana, Moura, Bruno, Reiser, Renata, Bedregal, Benjamin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2022
• Subjects: Artificial Intelligence; Computational Intelligence; Control; Engineering; Fuzzy Systems and Their Mathematics; Mathematical Logic and Foundations; Mechatronics; Robotics; Dimensional fuzzy sets; Cloud computing; Residuation property; Dimensional R-implications
• ISSN: 1432-7643; 1433-7479
• DOI: 10.1007/s00500-022-07221-6

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.