  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 M-natural-convexity" for functions defined
on a larger class of jump systems. By definition, every jump M-convex function
is a jump M-natural-convex function, and we show the equivalence of these
concepts by establishing an (injective) embedding of jump M-natural-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-natural-convex
functions admit a number of natural operations such as aggregation, projection
(partial minimization), convolution, composition, and transformation by a
network.
