Tropical linear algebra with the Łukasiewicz T-norm

Gavalec, Martin; Němcová, Zuzana; Sergeev, Sergĕı

doi:10.1016/j.fss.2014.11.008

Cited by 14 publications

(12 citation statements)

References 32 publications

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“…It can be seen from previous figures that the value 1 − λ divides the coordinate system into four parts. The eigenvectors can be described by way of the partition of indices of the vector components (introduced in [3]) according to their value. For Figure 5.…”

Section: The Equation (11) Holds If and Only If Eithermentioning

confidence: 99%

“…The complex algorithm for computation of the eigenspace using the (K, L) partition is given in [3], see Algorithm 3.1. With the knowledge, which of the partitions are secure, we can compute the generators of the tropical convex hull for every secure partition.…”

Section: The Eigenspace For Matrices Of Higher Dimensionsmentioning

confidence: 99%

“…The investigation of the eigenspace in max-prod algebra [7] is in preparation. Finally, papers [6,3] deals with Łukasiewicz fuzzy algebra.…”

Section: Introductionmentioning

confidence: 99%

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Steady states of max-Lukasiewicz fuzzy systems

Němcová¹

2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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The paper gives the systematic characterization of eigenspace in max-T algebra where T is equal to Łukasiewicz t-norm. Max-Łukasiewicz fuzzy algebra can be used for the description of the states of Discrete-event systems. The states can represent a balance of the resource unit expended during the evolution of the system (for example fuel or money). In the contribution, the classification of max-Łukasiewicz eigenspaces is described and explained by the two-dimensional examples -in this case it is possible to accompany the example with illustrative graphs. However the description of the eigenspace for the higher dimensions is also outlined.

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Section: The Equation (11) Holds If and Only If Eithermentioning

confidence: 99%

Section: The Eigenspace For Matrices Of Higher Dimensionsmentioning

confidence: 99%

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Steady states of max-Lukasiewicz fuzzy systems

Němcová¹

2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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“…The eigenvectors in a max-T algebra, for various triangular norms T, are useful in fuzzy set theory. Such eigenvectors have been studied in [17][18][19]. The eigenvalues and eigenvectors are interesting characteristics of the DES in fuzzy algebras.…”

Section: Introductionmentioning

confidence: 99%

“…The eigenspace structures for the drastic and t-norm have been studied in [18,19]. Finally, [17] describes the case of Łukasiewicz fuzzy algebra.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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.

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Steady states of max-Łukasiewicz fuzzy systems

Gavalec

Němcová

2017

Fuzzy Sets and Systems

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Tropical linear algebra with the Łukasiewicz T-norm

Cited by 14 publications

References 32 publications

Steady states of max-Lukasiewicz fuzzy systems

Steady states of max-Lukasiewicz fuzzy systems

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

Steady states of max-Łukasiewicz fuzzy systems

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