On the dimension of max-min convex sets

TL;DR

This paper introduces a new notion of dimension for max-min convex sets, establishing a connection with a max-min analogue of tropical rank and exploring its relation to strong regularity in max-min algebra.

Contribution

It defines a max-min convex set dimension and relates it to a novel max-min tropical rank, linking geometric and algebraic properties.

Findings

Max-min convex set dimension equals the max-min tropical rank.

The max-min tropical rank relates to strong regularity in max-min algebra.

The paper bridges geometric and algebraic concepts in max-min convexity.

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 trapezoidal property.