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-Therien hierarchy, using a characterization theorem for level 2 by Place and Zeitoun, we obtain that the level 2 is definable by omega-identities.