A note on Rogers semilattices of families of two embedded sets in the Ershov hierarchy

Abstract

We show that for every ordinal notation a of a successor ordinal > 1, there is a −1 a family A = {A,B} with A ⊂ B such that the Rogers semilattice of A has exactly one element. This extends a result of Badaev and Talasbaeva, proved for the case in which a is the ordinal notation of 2.