Decomposition orders—another generalisation of the fundamental theorem of arithmetic

Abstract

We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition order. We apply our result to establish that the commutative monoid of weakly normed processes modulo bisimulation definable in ACP ε with linear communication, with parallel composition as binary operation, has unique decomposition. We also apply our result to establish that the partial commutative monoid associated with a well-founded commutative residual algebra has unique decomposition.

A Finite Equational Base for CCS with Left Merge and Communication Merge

“…(The process 0 has as decomposition the empty multiset, and a parallel prime process p has as decomposition the singleton multiset [p].) The following theorem is a straightforward consequence of the main result in [10].…”

A finite equational base for CCS with left merge and communication merge

“…In a similar way as in [11,Sect. 4] it can be established that the inverse of → * is a decomposition order on the commutative monoid P γ with respect to parallel composition; it then follows from [11,Theorem 32] that this commutative monoid has unique decomposition.…”

“…(The process 0 has as decomposition the empty multiset, and a parallel prime process p has as decomposition the singleton multiset [p].) The following theorem is a straightforward consequence of the main result in [11].…”

On Unique Decomposition of Processes in the Applied π-Calculus

“…Luttik and van Oostrom [9] provided a generalization of the unique decomposition results for ordered monoids. They show that if the calculus satisfies certain properties, the unique decomposition result follows directly.…”