Rogers semilattices of limitwise monotonic numberings

Abstract

AbstractLimitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family is limitwise monotonic (l.m.) if every set is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice . The semilattices exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of ‐computable families. We show that every Rogers semilattice of a ‐computable family is isomorphic to some semilattice . On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices . In particular, there is an l.m. family S such that is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all ‐computable numberings. We prove that inside this class, the index set of l.m. numberings is ‐complete.