One-Element Rogers Semilattices in the Ershov Hierarchy

Abstract

The cardinality of the Rogers semilattice of a computable family of sets is its most natural invariant. The cardinality problem for semilattices of computable numberings, which was raised by Ershov [1] for families of computably enumerable (c.e.) sets, has long been canonical. It arises every time when more and more classes of families of objects become involved in the field of research.A fundamental answer to the question about the cardinality of Rogers semilattices for the classical case due to A. B. Khutoretskii [2] reads as follows: the semilattice of a computable family of c.e. sets is either one-element or infinite. An extensive bibliography on this subject is contained in [3][4][5][6]. All known results confirm Khutoretskii's dichotomy: the semilattice contains either at most one or infinitely many elements.And what is the reason for the above-mentioned dichotomy? Research in this direction was initiated by Mal'tsev [7] and Ershov [1], who tried to find a topological characterization of families with a one-element semilattice of computable numberings, and was continued in [8][9][10][11]. A complete * Supported by KN MON RK, grant No. AP08856834.

On the Existence of Universal Numberings

“…In view of the properties of the F. Stephan operator [17], it suffices to research Rogers semilattices for families of sets at two lower levels in the Ershov hierarchy. Other results on Rogers semilattices in Ershov hierarchy can be found, for example, in [18][19][20][21][22][23][24][25][26].…”