A note on M-convex functions on jump systems

A jump system is defined as a set of integer points (vectors) with a certain exchange property, generalizing the concepts of matroids, delta-matroids, and base polyhedra of integral polymatroids (or submodular systems). A discrete convexity concept is defined for functions on constant-parity jump systems and it has been used in graph theory and algebra. In this paper we call it "jump M-#-convexity" and extend it to "jump Mbconvexity" for functions defined on a larger class of jump systems. By definition, every jump M-convex function is a jump M-#-convex function, and we show the equivalence of these concepts by establishing an (injective) embedding of jump Mb-convex functions in n variables into the set of jump M-#-convex functions in n + 1 variables. Using this equivalence we show further that jump M-#-convex functions admit a number of natural operations such as aggregation, projection (partial minimization), convolution, composition, and transformation by a network. (C) 2020 Elsevier B.V. All rights reserved.