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I was thinking about the following some time ago. My question is whether such things have been studied before.

Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of dimension at most $n$ and relations (1) $P=Q$ if $P$ and $Q$ are congruent, (2) $P\cup Q=P+Q-P\cap Q$, (3) $\emptyset=0$.

Clearly the cokernel of the obvious map $E_{n-1}\to E_n$ is the usual $n$-th euclidean scissors congruence group.

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  • $\begingroup$ The answer is yes. The masters in this domain are Sah, Dupont, Cathelineau and Goncharov. You'll find a very precise and elegant analysis of the congruence groups and Dehn invariants in Scissors Congruences, Group Homology and Characteristic Classes by Dupont. Goncharov and Cathelineau are more on a way of generalizing this stuff to other fields of definitions and finding links with (Milnor) K-theory. $\endgroup$
    – few_reps
    Commented Oct 21, 2013 at 23:21
  • $\begingroup$ Does Dupont consider exactly this? Note that I am not declaring $P$ to be $0$ in $E_n$ if $P$ has dimension less than $n$. $\endgroup$
    – Tom Goodwillie
    Commented Oct 23, 2013 at 19:56
  • $\begingroup$ You're right, not exactly. I'd guess that the two points of view are equivalent, but it would need to be checked. Tell us ! $\endgroup$
    – few_reps
    Commented Oct 24, 2013 at 7:32

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For P=Q if P and Q are translates (instead of (1)), the so-called polytope algebra is obtained (see Peter McMullen, Advances 1989). The polytope algebra has found important applications in the theory of polytopes (McMullen 1993) and the theory of valuations on convex polytopes.

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