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

Abstract We are concerned with the $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -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 $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -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 $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -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 $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -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.