Tropical linear algebra with the Łukasiewicz T-norm

Martin Gavalec; Zuzana Němcová; Sergey Sergeev

doi:10.1016/j.fss.2014.11.008

Tropical linear algebra with the Łukasiewicz T-norm

Martin Gavalec, Zuzana Němcová, Sergey Sergeev

Mathematics

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8 Citations (Scopus)

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Abstract

The max-Łukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics “a+b”=max⁡(a,b) and “ab”=max⁡(0,a+b−1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Łukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Łukasiewicz semiring.

Original language	English
Pages (from-to)	131-148
Journal	Fuzzy Sets and Systems
Volume	276
Early online date	12 Nov 2014
DOIs	https://doi.org/10.1016/j.fss.2014.11.008
Publication status	Published - 1 Oct 2015

Keywords

Matrix power
Tropical
Max-plus
Łukasiewicz
Eigenvector

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10.1016/j.fss.2014.11.008

Gavalec_Nemcova_Sergeev_Tropical_linear_algebra_Fuzzy_Sets_Systems_2014
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http://linkinghub.elsevier.com/retrieve/pii/S0165011414005119

Perron-Frobenius Theory and Max-Algebraic Combinatorics of Nonnegative Matrices
Butkovic, P.
Engineering & Physical Science Research Council
12/03/12 → 11/03/14
Project: Research Councils

Cite this

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title = "Tropical linear algebra with the {\L}ukasiewicz T-norm",

abstract = "The max-{\L}ukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics “a+b”=max⁡(a,b) and “ab”=max⁡(0,a+b−1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-{\L}ukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-{\L}ukasiewicz semiring.",

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author = "Martin Gavalec and Zuzana N{\v e}mcov{\'a} and Sergey Sergeev",

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doi = "10.1016/j.fss.2014.11.008",

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AU - Gavalec, Martin

AU - Němcová, Zuzana

AU - Sergeev, Sergey

PY - 2015/10/1

Y1 - 2015/10/1

N2 - The max-Łukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics “a+b”=max⁡(a,b) and “ab”=max⁡(0,a+b−1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Łukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Łukasiewicz semiring.

AB - The max-Łukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics “a+b”=max⁡(a,b) and “ab”=max⁡(0,a+b−1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Łukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Łukasiewicz semiring.

KW - Matrix power

KW - Tropical

KW - Max-plus

KW - Łukasiewicz

KW - Eigenvector

U2 - 10.1016/j.fss.2014.11.008

DO - 10.1016/j.fss.2014.11.008

M3 - Article

SN - 0165-0114

VL - 276

SP - 131

EP - 148

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

ER -

Tropical linear algebra with the Łukasiewicz T-norm

Abstract

Keywords

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Perron-Frobenius Theory and Max-Algebraic Combinatorics of Nonnegative Matrices

Cite this