An $ omega $-complete equational specification of interleaving

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.