Rogers semilattices of limitwise monotonic numberings

Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family (Formula presented.) is limitwise monotonic (l.m.) if every set (Formula presented.) is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice (Formula presented.). The semilattices (Formula presented.) exhibit a peculiar behavior, which puts them in-between the classical Rogers semilattices (for computable families) and Rogers semilattices of (Formula presented.) -computable families. We show that every Rogers semilattice of a (Formula presented.) -computable family is isomorphic to some semilattice (Formula presented.). On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices (Formula presented.). In particular, there is an l.m. family S such that (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) -computable numberings. We prove that inside this class, the index set of l.m. numberings is (Formula presented.) -complete.