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Linear independence over tropical semirings and beyond
The symmetrization of the max-plus algebra is revisited, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max- plus algebra and developing some general technique to prove combinatorial and polynomial identities for matricesover semirings.
Tropical Polyhedra are Equivalent to mean Payoff Games
It is shown that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems, and obtained as a corollary a game theoretical proof of the fact that the tropical rank of a matrix coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense.
RANK INEQUALITIES OVER SEMIRINGS
Inequalities on the rank of the sum and the product of two matrices over semirings are surveyed. Preferences are given to the factor rank, row and column ranks, term rank, and zero-term rank of…
Tropical Cramer Determinants Revisited
We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or…
Linear Preservers of Extremes of Rank Inequalities over Semirings: The Factor Rank
We characterize linear preservers for sets of matrix ordered tuples which satisfy extremal properties with respect to factor rank.
The correspondence between tropical convexity and mean payoff games
We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum, two player game problems. In particular, we set up an equivalence between the…
Majorization for matrix classes
Commutative matrix subalgebras and length function