Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in $V$ with multiplication induced by Minkowski sum.

Let $P$ be a convex polytope in $V$, and consider the subring $A(P)$ of $L(V)$ generated by the faces of $P$ (including $P$ itself).

Question: Has $A(P)$ been studied at all?

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    $\begingroup$ Beside (and probably before) Morelli the "big" algebraic structure was studied by Peter McMullen (algebra of polytopes) and Pukhlikov - Khovanskii (virtual polytopes). $\endgroup$ – Ivan Izmestiev Nov 20 '18 at 18:32

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