  Mazurkiewicz traces describe concurrent behaviors of distributed systems.
Trace-closed word languages, which are "linearizations" of trace languages,
constitute a weaker notion of concurrency but still give us tools to
investigate the latter. In this vein, our contribution is twofold. Firstly, we
develop definitions that allow classification of $\omega$-regular trace
languages in terms of the corresponding trace-closed $\omega$-regular word
languages, capturing E-recognizable (reachability) and (deterministically)
B\"uchi recognizable languages. Secondly, we demonstrate the first
automata-theoretic result that shows the equivalence of $\omega$-regular
trace-closed word languages and Boolean combinations of deterministically
$I$-diamond B\"uchi recognizable trace-closed languages.
