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Bases in max-algebra. (English) Zbl 1059.15001

The authors consider bases for spaces of vectors over the semiring \((\mathbb R, \max, +)\). It is proved that all bases for a subset have the same cardinality; there exist subsets of \(\mathbb R^n\) having bases of arbitrary cardinality iff \(n\geq 3\); subsets have (finite) bases iff they are finitely generated, that \(\mathbb R^n\), \(n>1\) has no basis in this sense, and if bases were allowed to be infinite, under certain conditions bases would still not exist. One special feature of the definition is that in writing \(w\) as a linear combination of some set \(U\) of vectors, the linear combination is not allowed to involve \(w\) itself.

MSC:

15A80 Max-plus and related algebras
15A03 Vector spaces, linear dependence, rank, lineability
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