  Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex
function and M${}^\natural$-convex function as discrete convex functions, which
are generalizations of valuated matroids due to Dress and Wenzel (1992). In the
present paper we consider a new operation defined by a convolution of sections
of an M${}^\natural$-convex function that transforms the given
M${}^\natural$-convex function to an M-convex function, which we call a
compression of an M${}^\natural$-convex function. For the class of valuated
generalized matroids, which are special M${}^\natural$-convex functions, the
compression induces a valuated permutohedron together with a decomposition of
the valuated generalized matroid into flag-matroid strips, each corresponding
to a maximal linearity domain of the induced valuated permutohedron. We examine
the details of the structure of flag-matroid strips and the induced valuated
permutohedron by means of discrete convex analysis of Murota.
