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 $S\subset P\left(\omega \right)$ is limitwise monotonic (l.m.) if every set $\nu \left(k\right)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice $R_{lm}\left(S\right)$. The semilattices $R_{lm}\left(S\right)$ exhibit a peculiar behavior, which puts them in-between the classical Rogers semilattices (for computable families) and Rogers semilattices of ${\Sigma }_2^0$-computable families. We show that every Rogers semilattice of a ${\Sigma }_2^0$-computable family is isomorphic to some semilattice $R_{lm}\left(S\right)$. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices $R_{lm}\left(S\right)$. In particular, there is an l.m. family S such that $R_{lm}\left(S\right)$ 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 $R_{lm}\left(S\right)$ 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 ${\Sigma }_2^0$-computable numberings. We prove that inside this class, the index set of l.m. numberings is ${\Sigma }_4^0$-complete.