We deal with the question of the $\omega$-reducibility of pseudovarieties of ordered monoids. A pseudovariety of ordered monoids $\mathsf{V}$ is called $\omega$ reducible if, given a finite monoid $M$, for every inequality of pseudowords that is valid in $\mathsf{V}$, there exists an inequality of $\omega$-words that is also valid in $\mathsf{V}$ and has the same “imprint” in $M$. In other words, we investigate the $\omega$-reducibility of an “ordered version” of $\mathsf{V}$-pointlike pairs, where $\mathsf{V}$ is a pseudovariety of ordered monoids.

Specifically, we are interested in pseudovarieties $\mathsf{V}$ representing half levels of concatenation hierarchies of regular languages. In a concatenation hierarchy, the $\omega$-reducibility of a half level implies the definability of the next half level by $\omega$-inequalities. We will discuss how to prove the $\omega$-reducibility of levels 1/2 and 3/2 of a concatenation hierarchy with a locally finite basic pseudovariety. One way is to use recent results on the covering problem for regular languages by Thomas Place and Marc Zeitoun. Another way is to prove the $\omega$-reducibility directly by manipulating inequalities of words valid in appropriate (locally finite) pseudovarieties (constituing a “stratification” of a given pseudovariety). The latter technique has a potential to be generalized to higher levels of concatenation hierarchies. Such a generalization is the work in progress.

This talk is based primarily on my recently defended Ph.D. thesis at Masaryk University.