Remarks on Piecewise-linear Algebra
نویسنده
چکیده
Introduction* A function f: V —>W between real vector spaces is piecewise-linear (PL) if there exists a partition of V into "open polyhedra" X€ (i.e., relative interiors of polyhedra) such that / is affine on each Xt. (As distinct to the case of PL-topology, no continuity is required of /.) Images and preimages under PL-maps give rise to finite unions of open polyhedra, or PL-seίs; conversely PL functions can be characterized by the fact that their graphs are PL-sets. This paper studies some basic algebraic properties of the category PL, proving in particular that it is an exact category, and in fact a pretopos. A classification is given for the isomorphism classes of objects of PL, in terms of a two-generator semiring.
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