Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-?ukasiewicz Algebra

Martin Gavalec, Zuzana Němcová and Ján Plavka
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Martin Gavalec: Faculty of Informatics and Management, University of Hradec Králové, 50003 Hradec Králové, Czech Republic
Zuzana Němcová: Faculty of Informatics and Management, University of Hradec Králové, 50003 Hradec Králové, Czech Republic
Ján Plavka: Faculty of Electrical Engineering and Informatics, Technical University of Košice, 04200 Košice, Slovakia

Mathematics, 2020, vol. 8, issue 9, 1-19

Abstract: The ?ukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the ?ukasiewicz t-norm forms the basis for the so-called max-?uk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max- T algebra (with a given t-norm, T ). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-?uk algebra. The main method used in this paper is based on max-? linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

Keywords: max-?ukasiewicz algebra; interval matrix; interval eigenvector; strong interval eigenvector (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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