In C-H Sah's book Hilbert's third problem: scissors congruence, the author defines the data for abstract scissors congruence in order to prove Zylev's theorem by combinatorial means in great abstraction. I expect that most of the book is over my head as an undergrad but I'm looking for clarification as to wether the definition Sah gives is actually correct. My advisor and I could not figure out how to resolve the apparent discrepancies.
The definition is given as follows on page 5:
The abstract scissors congruence data consists of a distinguished family of nonempty subsets (to be called $n$-simplices where $n$ is to be interpreted as the dimension) of a nonempty set $X$ and a specified equivalence relation (called congruence) among the $n$-simplices. We need a few definitions before stating the addition condition to be satisfied by the $n$-simplices.
Two $n$-simplices $A$ and $B$ are said to be interior disjoint if the following conditions hold:
- (D1) $A \cap B$ contains no $n$-simplices, and
- (D2) if $C$ is an $n$-simplex contained in $A \cup B$, then $C \subset A$ if and only if $C \cap B$ contains no $n$-simplices.
A polyhedron $P$ is understood to be a finite pairwise interior disjoint union of $n$-simplices. The concept of interior disjoint union can then be extended to polyhedra with $n$-simplices replaced by nonempty polyhedra. We will use $\coprod$ to denote pairwise interior disjoint unions (as well as direct sum when there is no chance of confusion). We omit the proof of the following elementary result:
Lemma 2.1. If $P$, $Q$ and $R$ are polyhedra with $P \coprod R = Q \coprod R$, then $P = Q$.
If $A$, $B$, $C$ are $n$-simplices with $A = B \coprod C$, we say $A$ is simply subdivided into $B$ and $C$. If $P = \coprod P_i$ is a polyhedron where each $P_i$ is an $n$-simplex, then a subdivision of $P$ is understood to be a finite succession of simple subdivisions such that each simple subdivision is performed on one of the $n$-simplices exhibited in the preceeding step. For example, the first step may be a simple subdivision of $P_1$ into $Q$ and $R$; the second step may be a simple subdivision of $Q$ or $R$ or any $P_j$ with $j > 1$; and so on.
To complete the abstract scissors congruence data, we impose the following condition:
- (S) Let $A$ and $B$ be $n$-simplices. Then there is at least one subdivision of $A$, say $A = \coprod_{1 \le i \le t} A_i$, such that $A \cap B = \coprod_{j \in J} A_j$ for some $J \subset \{ 1,\dots,t \}$.
My issue is with the axiom (S). It seems that the expression $A \cap B = \coprod_{j \in J} A_j$ is either not well-defined or implicitly states the idea that $A \cap B$ must be a polyhedron. Taking the second interpretation, if we consider the usual case of Euclidean scissors congruence then this axiom does not hold, as the intersection of two triangles joined side to side is not a polygon.
Another example is given on the next page: for $n \ge 0$, let the $n$-simplices be arbitrary $(n+1)$-element subsets of an infinite set $X$, and congruence is equinumerosity. If $n = 0$ then the polyhedra are just finite sets and interior disjointness is set-theoretic disjointness. Sah then states that if $n>0$ then no $n$-simplices are interior disjoint (because D2 never holds) and moreover he states that '(S) is clear and cardinality is a complete invariant'. As far as I can tell (S) does not actually hold in this case either, as the intersection of two $(n+1)$-element sets may have fewer elements but not be empty.
Keeping the Euclidean simplices in mind, the most obvious way to fix the statement of axiom (S) would be to require $A$ and $B$ to either be interior disjoint or else satisfy the current definition of (S). This still does not fix the finite subset case, though, as no two simplices are interior disjoint.
Presumably the entire book is not built on nonsense, so what is the proper definition used by Sah and other authors that refer to this book? Another definition in terms of category theory (specifically the idea of double categories) is given by Inna Zakharevich in her thesis Scissors Congruence and K-Theory, but it strikes me as a bit more complicated, and I was hoping for a more elementary definition requiring less heavy machinery.
