On the dimension of max–min convex sets

Viorel Nitica; Sergey Sergeev

doi:10.1016/j.fss.2014.10.008

On the dimension of max–min convex sets

Viorel Nitica, Sergey Sergeev

Mathematics

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Abstract

We introduce a notion of dimension of max–min convex sets, following the approach of tropical convexity. We introduce a max–min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max–min algebra, which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property.

Original language	English
Pages (from-to)	88-101
Journal	Fuzzy Sets and Systems
Volume	271
Early online date	18 Oct 2014
DOIs	https://doi.org/10.1016/j.fss.2014.10.008
Publication status	Published - 15 Jul 2015

Keywords

Max–min algebra
Dimension
Tropical convexity
Tropical rank
Strongly regular matrix

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10.1016/j.fss.2014.10.008

Nitica_Sergeev_On_dimension_max_min_convex_sets_Fuzzy_Sets_Systems_2014
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http://linkinghub.elsevier.com/retrieve/pii/S0165011414004552

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 = "On the dimension of max–min convex sets",

abstract = "We introduce a notion of dimension of max–min convex sets, following the approach of tropical convexity. We introduce a max–min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max–min algebra, which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property.",

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AU - Sergeev, Sergey

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N2 - We introduce a notion of dimension of max–min convex sets, following the approach of tropical convexity. We introduce a max–min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max–min algebra, which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property.

AB - We introduce a notion of dimension of max–min convex sets, following the approach of tropical convexity. We introduce a max–min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max–min algebra, which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property.

KW - Max–min algebra

KW - Dimension

KW - Tropical convexity

KW - Tropical rank

KW - Strongly regular matrix

U2 - 10.1016/j.fss.2014.10.008

DO - 10.1016/j.fss.2014.10.008

M3 - Article

SN - 0165-0114

VL - 271

SP - 88

EP - 101

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

ER -

On the dimension of max–min convex sets

Abstract

Keywords

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Projects

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

Cite this