Allgemeine Zerlegungstheorie. (General decomposition theory) (Q1198222)

In the present paper, a common generalization to the following two types of decomposing a set is given: the decomposition of polyhedra in the sense of elementary geometry (see, e.g., the literature on scissors congruences) and the disjoint decomposition of sets [cf. \textit{S. Wagon}, The Banach-Tarski paradox (1985; Zbl 0569.43001)]. This generalization is called the \(M\)-disjoint decomposition of a set \(C\). Let \(R\neq\emptyset\) be a set, \(G\) a subgroup of the symmetric group \(S_ R\) and \(M\subseteq2^ R\) with \(\emptyset\in M\) and \(| M|>1\); then \(A,B\subseteq2^ R\) are called \(M\)-disjoint \((A\square B=\emptyset)\) if \(\emptyset\) is the only set of \(M\) contained in \(A\cap B\). For \(A,B,C\subseteq R\), the set \(C\) denotes the \(M\)-disjoint union of \(A\) and \(B\) (\(C=A\blacksquare B\)) if \(C=A\cup B\) and \(A\square B=\emptyset\). The relation of decomposition equivalence, obtained on the base of the notions above, shows common properties of both areas of decomposition theory. The general properties of this relation are given, too.