Discrete piecewise linear functions

TL;DR

This paper introduces permutographs and characterizes integral functions on them, providing lattice polynomial representations for piecewise linear functions on convex domains and continuous selectors on linear orders.

Contribution

It introduces permutographs and characterizes integral functions, linking them to lattice polynomial representations of piecewise linear functions and selectors.

Findings

Characterization of integral functions on permutographs.

Representation of piecewise linear functions as lattice polynomials.

Representation of continuous selectors on linear orders.

Abstract

The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is used to establish lattice polynomial representations of piecewise linear functions on convex domains and continuous selectors on linear orders.