  Let X and Y be finite dimensional normed spaces, F(X,Y) a collection of all
mappings from X into Y. A mapping $P\in F(X,Y)$ is said to be piecewise affine
if there exists a finite family of convex polyhedral subsets covering X and
such that the restriction of $P$ on each subset of this family is an affine
mapping. In the paper we prove a number of characterizations of piecewise
affine mappings. In particular we show that a mapping $P: X \to Y$ is piecewise
affine if and only if for any partial order $\preceq$ defined on Y by a
polyhedral convex cone both the $\preceq$-epigraph and the $\preceq$-hypograph
of P can be represented as the union of finitely many convex polyhedral subsets
of $X \times Y$. When the space Y is ordered by a minihedral cone or
equivalently when Y is a vector lattice the collection F(X,Y) endowed with
standard pointwise algebraic operations and the pointwise ordering is a vector
lattice too. In the paper we show that the collection of piecewise affine
mappings coincides with the smallest vector sublattice of F(X,Y) containing all
affine mappings. Moreover we prove that each convex (with respect to an
ordering of Y by a minihedral cone) piecewise affine mapping is the least upper
bound of finitely many affine mappings. The collection of all convex piecewise
affine mappings is a convex cone in F(X,Y) the linear envelope of which
coincides with the vector subspace of all piecewise affine mappings.
