Tropical matrix duality and Green's 𝔇 relation

Hollings, Christopher; Kambites, Mark

doi:10.1112/jlms/jds015

Cited by 24 publications

(51 citation statements)

References 27 publications

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“…In fact, there is a correspondence between inclusions of row spaces and certain surjections of column spaces; see [8] for full details. to R(B) taking the ith row of B to the ith row of A for all i.…”

Section: Green's Relations Idempotents and Regularitymentioning

confidence: 99%

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Tropical matrix groups

2017

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Abstract. We study the subgroup structure of the semigroup of real square matrices of given dimension under tropical matrix multiplication. We show that every maximal subgroup is isomorphic to the full linear automorphism group of a related tropical polytope, and that each of these groups is the direct product of R with a finite group. We also show that there is a natural and canonical embedding of each full rank maximal subgroup into the group of units of the semigroup. Out results have numerous corollaries, including the fact that every automorphism of a full rank projective tropical polytope extends to an automorphism of the containing space, and that every full rank subgroup has a common eigenvector.

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Section: Green's Relations Idempotents and Regularitymentioning

confidence: 99%

“…In [8], Hollings and the third author gave a complete description of the D-relation for M n (FT), using the duality between the row and column space of a tropical matrix. In [11] the second and third authors described the equivalence relation J and pre-order ≤ J in in M n (FT) and M n (T).…”

Section: Green's Relations Idempotents and Regularitymentioning

confidence: 99%

Tropical matrix groups

2017

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“…This operation is a residuation operator in the sense of residuation theory [2], and has been extensively applied in max-plus algebra and geometry (see for example [1,5,8,9]). We record a number of useful properties, which the reader can easily verify using the definition above:…”

Section: Residuation and Dominationmentioning

confidence: 99%

“…The maps ρ A and χ A are mutually inverse bijections between the max-plus row space and the max-plus column space of the matrix A; they have many interesting properties -see for example [5,7,9]. When the matrix A is a Kleene star, it turns out that the duality maps have a particularly nice form:…”

Section: Kleene Stars and Dominator Matricesmentioning

confidence: 99%

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Convexity of tropical polytopes

Johnson

Kambites

2015

Linear Algebra and its Applications

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Abstract. We study the relationship between min-plus, max-plus and Euclidean convexity for subsets of R n . We introduce a construction which associates to any max-plus convex set with compact projectivisation a canonical matrix called its dominator. The dominator is a Kleene star whose max-plus column space is the min-plus convex hull of the original set. We apply this to show that a set which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii) a Euclidean polytope must also be the third. In particular, these results answer a question of Sergeev, Schneider and Butkovič [15] and show that row spaces of tropical Kleene star matrices are exactly the "polytropes" studied by Joswig and Kulas [12].The notion of tropical convexity (also known as max-plus or min-plus convexity) has long played an important role in max-plus algebra and its numerous application areas (see for example [3,4]). More recently, applications have emerged in areas of pure mathematics as diverse as algebraic geometry [7] and semigroup theory [9,10].Recall that a subset of R n is called max-plus convex if it is closed under the operations of componentwise maximum ("max-plus sum") and of adding a fixed value to each component ("tropical scaling"). A max-plus polytope is a non-empty max-plus convex set which is generated under these operations by finitely many of its elements; max-plus polytopes are exactly the row spaces (or column spaces) of matrices over the max-plus semiring. There are obvious dual notions of min-plus convexity and min-plus polytopes. Minplus and max-plus polytopes are sometimes called tropical polytopes 3 . The projectivisation of a max-plus polytope is the set of orbits of its points under the action of tropical scaling. Since any point can be scaled to put 0 in the first coordinate (say), restricting the polytope to points with first coordinate 0 gives a cross-section of the scaling orbits, and hence a natural identification of the projectivisation with a subset of R n−1 . A subset which arises from a max-plus polytope in this way we term a projective max-plus polytope. In general, a projective max-plus polytope is a compact Euclidean polyhedral complex in R n−1 ; it may or may not be a convex set in the ordinary Euclidean metric. Joswig and Kulas [12] studied the class 1 Email Marianne.Johnson@maths.manchester.ac.uk. 2 Email Mark.Kambites@manchester.ac.uk. 3 Typically one fixes upon either the "min convention" or the "max convention" and uses terms such as "tropically convex" and "tropical polytope" to refer to min-plus or max-plus as appropriate. A key feature of this paper is that we study the relationship between min-plus and max-plus convexity, so for the avoidance of ambiguity we will tend not to use the word "tropical".

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Polytropes and Tropical Eigenspaces: Cones of Linearity

Tran

2014

Discrete Comput Geom

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The map which takes a square matrix A to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of R n×n , whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of R n×n corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels [13] and lead to a new combinatorial classification of polytropes and tropical eigenspaces.

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Tropical matrix duality and Green's 𝔇 relation

Cited by 24 publications

References 27 publications

Tropical matrix groups

Tropical matrix groups

Convexity of tropical polytopes

Polytropes and Tropical Eigenspaces: Cones of Linearity

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