Contributions to max–min convex geometry. I: Segments Original Research Article

We give some contributions to the theory of “max–min convex geometry”, that is, convex geometry in the semimodule image over the max-min semiring Rmax,min=Runion or logical sum{-∞,+∞}. We introduce “elementary segments” that generalize from n=2 the horizontal, vertical or oblique segments contained in the main bisector of image. We show that every segment in image 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”.