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.
Highlights
The max–min semiring is defined as the unit interval B = [0, 1] with the operations a ⊕ b := max(a, b), as addition, and a ⊗ b := min(a, b), as multiplication
In the remaining part of the paper, following the parallel with the tropical rank considered by Develin, Santos and Sturmfels [7] in the max–plus algebra, we investigate how our notion of dimension relates with the notion of strong regularity in max–min algebra
We show that for A ∈ B(k, k) our notion of strong regularity is equivalent to the trapezoidal property, and it coincides with the strong regularity in max–min algebra introduced in [2]
Summary
The present paper aims to develop a new geometric approach to the well-known notions of strong regularity and matrix rank in max–min algebra. To this end, it seems to be the first paper that connects max–min linear algebra with max–min convexity. Of the possibilistic measures π1, π2 with possibilities α, β, max(α, β) = 1, is defined as max(min(α, π1), min(β, π2)), that is, as a point on the max–min segment [π1, π2] This is a particular case of extended mixtures of decomposable measures (which are a family of set functions encompassing probability measures and necessity and possibility measures as particular cases), as studied in [9] where the application to utility theory is pointed out.
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