Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.

A real finite-dimensional vector space $V$ defines the following two spaces, also real. $\mathcal{P}(V)$ generated by indicator functions of closed convex polyhedra in $V$ (not necessarily bounded) and $\mathcal{C}_b(V)$ generated by indicator functions of convex compacts in $V$. Let then $$\mathcal{P}_b(V)=\mathcal{P}(V)\cap\mathcal{C}_b(V).$$ This seems to obviously be generated by indicators of bounded closed polyhedra but, just in case, I present the definitions as in the book.

Each of these three spaces can be equipped with a nontrivial multiplicative structure by setting $$\chi_X*\chi_Y=\chi_{X+Y}$$ (Minkowski sum) for two generating indicator functions $\chi_X$ and $\chi_Y$. (That this is a well-defined algebra structure is far from obvious but is shown in the book.)

Now, problem 10 in chapter 3 states: $\mathcal{P_b}(V)$ has no zero-divisors with respect to $*$.

This seems to be wrong in dimension 1:$$(\chi_{[0;1]}-\chi_{\{0\}})*(\chi_{[0;1]}-\chi_{\{1\}})=\chi_{[0;2]}-\chi_{[0;1]}-\chi_{[1;2]}+\chi_{\{1\}}=0.$$

So... Am I missing something? If not, I would suppose that there exists some correct version which was supposed to be here instead. Any idea what that could look like?