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 ACPe 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

Ordered semigroups and monoids (msc 06F05), Partial algebras (msc 08A55), Applications of universal algebra in computer science (msc 08A70), Multiplicative structure; Euclidean algorithm; greatest common divisors (msc 11A05), Factorization; primality (msc 11A51), Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) (msc 68Q85)
Software Engineering [SEN]
Specification and Analysis of Embedded Systems

Luttik, S.P, & van Oostrom, V. (2004). Decomposition orders : another generalisation of the fundamental theorem of arithmetic. Software Engineering [SEN]. CWI.