A purely algebraic proof of the omega-reducibility of pseudovarieties representing low half levels of concatenation hierarchies

Abstract

We are concerned with the ω-reducibility of pseudovarieties of ordered monoids representing half levels of concatenation hierarchies. In the author’s paper (Int. J. Algebra Comput. 64(01), 87–135, 2024), the ω-reducibility of pseudovarieties representing levels 1/2 and 3/2 of concatenation hierarchies with a locally finite basic pseudovariety has been proven, using results of the paper by Place (Log. Methods Comput. Sci. 14(4:16), 1–58, 2018) on so called covering of corresponding sets of regular languages. In this paper, we prove the same results on the ω-reducibility, not using the results of the mentioned paper by Place, although still inspired by their proofs. This new method of the proofs of the ω-reducibility prepares us to their potential extension to higher half levels of concatenation hierarchies. The process of a gradual generalization is initiated in this paper.