The omega-reducibility of pseudovarieties of ordered monoids representing low levels of concatenation hierarchies

We deal with the question of the $\omega$-reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids $V$ is called $\omega$-reducible if, given a finite ordered monoid M, for every inequality of pseudowords that is valid in $V$, there exists an inequality of $\omega$-words that is also valid in $V$ and has the same “imprint” in M.

Place and Zeitoun have recently proven the decidability of the membership problem for levels $1∕2$, 1, $3∕2$ and $5∕2$ of concatenation hierarchies with level 0 being a finite Boolean algebra of regular languages closed under quotients. The solutions of these membership problems have been found by considering a more general problem of separation of regular languages and its further generalization — a problem of covering. Following the results of Place and Zeitoun, we prove that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels $1∕2$ and $3∕2$ are $\omega$-reducible. As a corollary of these results, we obtain that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels $3∕2$ and $5∕2$ are definable by $\omega$-inequalities. Furthermore, in the special case of the Straubing–Thérien hierarchy, using a characterization theorem for level 2 by Place and Zeitoun, we obtain that the level 2 is definable by $\omega$-identities.