Abstract dependence, recursion theory, and the lattice of recursively enumerable filters
Rodney G. Downey · Bulletin of the Australian Mathematical Society · 1983
This thesis is divided into two sections.Both sections are devoted to the study of effectiveness in algebra, realized as analyses of substructures of recursive structures.Section One deals with closed subsets of a Steinitz closure system with recursive dependence as introduced by Metakides and Nerode.Initially we generalize many results proved by Metakides and Nerode and results concerning LiV^J) , the lattice of recursively enumerable subspaces, to considerably more general settings.To do this we introduce the notions of semiregularity and the closure intersection property, and show how they account for most of the observed phenomena in i(^0 O ) and L(u) , the lattice of recursively enumerable sets.For example, we show that if (#, cl) has the closure intersection property and is semiregular then Th (£(£/)) is undecidable.Similarly, we show that as (F m t cl) is regular, the Karp-%hill theorem fails for Shore defined nowhere simplicity in L(u>) .We examine analogues of this notion in L(U) , and in particular L(V m ) .If {U, cl) has the closure intersection property then any recursively enumerable nondecidable closed subset can be decomposed into a pair of recursively enumerable nondecidable nowhere simple closed subsets.We use this for results concerning automorphisms of i(f