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We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux.… (More)
— 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 external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action… (More)
We study non-linear surjective mappings on subsets of M n (F), which preserve the zeros of some fixed polynomials in noncommuting variables.
Idempotent matrices play a significant role while dealing with different questions in matrix theory and its applications. It is easy to see that over a field any idempotent matrix is similar to a diagonal matrix with 0 and 1 on the main diagonal. Over a semiring the situation is quite different. For example, the matrix J of all ones is idempotent over… (More)
In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the algebras generated by two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be… (More)