Contributions to max–min convex geometry. I: Segments

We give some contributions to the theory of “max–min convex geometry”, that is, convex geometry in the semimodule over the max-min semiring Rmax,min=R∪{-∞,+∞}. We introduce “elementary segments” that generalize from n=2 the horizontal, vertical or oblique segments contained in the main bisector of . We show that every segment in is a concatenation of a finite number of elementary subsegments (at most 2n-1, respectively at most 2n-2, in the case of comparable, respectively, incomparable, endpoints x,y). In this first part we study “max–min segments”, and in the subsequent second part (submitted) we study “max–min semispaces” and some of their relations to “max–min convex sets”.