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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.

We present an axiomatic approach to meaninglessness in finite and transfinite term rewriting and lambda calculus. We justify our axioms in two ways. First, they are shown to imply important properties of meaninglessness: genericity of the class of meaningless terms, the consistency of equating all meaningless terms, and the construction of Böhm trees. Second we show that they can be easily verified for existing notions of meaninglessness

A graph language L is in the class C-edNCE of context-free edNCE graph languages if and only if L= f (T ) where f is a partial function on graphs that can be defined in monadic second-order logic and T is the set of all trees over some ranked alphabet. This logical characterization implies a large number of closure and decidability properties of the context-free edNCE graph languages. Rather than context-free graph grammars we use regular path descriptions to define graph languages.] 1997 Academic Press