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

Abstract

We deal with the question of the [Formula: see text]-reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids [Formula: see text] is called [Formula: see text]-reducible if, given a finite ordered monoid M, for every inequality of pseudowords that is valid in [Formula: see text], there exists an inequality of [Formula: see text]-words that is also valid in [Formula: see text] and has the same “imprint” in M. Place and Zeitoun have recently proven the decidability of the membership problem for levels [Formula: see text], 1, [Formula: see text] and [Formula: see text] 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 [Formula: see text] and [Formula: see text] are [Formula: see text]-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 [Formula: see text] and [Formula: see text] are definable by [Formula: see text]-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 [Formula: see text]-identities.