Rogers semilattices of limitwise monotonic numberings

Abstract

Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S⊂Pfalse(ωfalse)$S\subset P(\omega )$ is limitwise monotonic (l.m.) if every set νfalse(kfalse)$\nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice Rlm(S)$R_{lm}(S)$. The semilattices Rlm(S)$R_{lm}(S)$ exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of normalΣ20$\Sigma ^0_2$‐computable families. We show that every Rogers semilattice of a normalΣ20$\Sigma ^0_2$‐computable family is isomorphic to some semilattice Rlm(S)$R_{lm}(S)$. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices Rlm(S)$R_{lm}(S)$. In particular, there is an l.m. family S such that Rlm(S)$R_{lm}(S)$ 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 Rlm(S)$R_{lm}(S)$ 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 normalΣ20$\Sigma ^0_2$‐computable numberings. We prove that inside this class, the index set of l.m. numberings is normalΣ40$\Sigma ^0_4$‐complete.