Allgemeine Zerlegungstheorie

Abstract

The notion “ℳ-disjoint decomposition” is introduced as a common generalization of the “elementary geometric decomposition of polyhedras” and the “disjoint decomposition of sets”: Let R≠ø be a set, G a subgroup of the symmetric group S_R and ℳ$\subseteq$ 2^R with ø∈ℳ and ¦ℳ¦>1; then A, B$\subseteq$ 2^R are called ℳ-disjoint if ø. is the only set of ℳ contained in A ∩ B, and then A ▪ B ≔ A ∪ B is called the disjoint union. If ℰ$\subseteq$ ℳ fullfills the axioms (0, I, II, III) of §2, then ℰ can be provided in a natural way with certain binary relations. With respect to the decomposition equivalence

can be turned in a partially ordered commutative semigroup

with identity (cf. Satz §5).