Prove the general enrichment link between the ST and DD temporal hierarchies at all levels

Prove that for every integer n ≥ 1, the nth level in the temporal hierarchy with basis DD (i.e., nDD) equals the enrichment (wreath product) of the nth level in the temporal hierarchy with basis ST by the class of suffix languages SUF (the Boolean algebra generated by languages of the form A* w with w ∈ A*); equivalently, establish nDD = nST ◦ SUF for all n ≥ 1.

Background

The paper studies temporal hierarchies of regular languages built via an operator that adds future/past modalities parameterized by a basis class. Two important bases are ST = {∅, A*} and DD = ST^+, which correspond to standard unary temporal logics and are closely related to the classic Straubing–Thérien and dot-depth concatenation hierarchies.

Using results on enrichment (also known as the wreath product of language classes), the authors show a specific connection at level two: {2}{DD} = {2}{ST} ◦ SUF, where SUF denotes the class of suffix languages (Boolean combinations of sets A* w). In the conclusion, they assert that this connection should generalize to all levels n, but they do not provide a proof and explicitly defer it to future work.

The open task is to supply a proof of this general statement, thereby establishing a uniform relationship between the temporal hierarchies of bases ST and DD through enrichment by SUF for every level.