We consider the process theory $PA$ that includes an operation for parallel composition, based on the interleaving paradigm. We prove that the standard set of axioms of $PA$ is not $omega$-complete by providing a set of axioms that are valid in $PA$, but not derivable from the standard ones. We prove that extending $PA$ with this set yields an $omega$-complete specification, which is finite in a setting with finitely many actions.

Concurrent Programming (acm D.1.3), Models of Computation (acm F.1.1), Modes of Computation (acm F.1.2)
Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) (msc 68Q10), Abstract data types; algebraic specification (msc 68Q65), Algebraic theory of languages and automata (msc 68Q70)
Software (theme 1)
Software Engineering [SEN]
Software Analysis and Transformation

Fokkink, W.J, & Luttik, S.P. (2000). An $ omega $-complete equational specification of interleaving. Software Engineering [SEN]. CWI.