In this paper, we study the lattice of r.e. subspaces of a recursively presented vector space V ∞ with regard to the various complexity-theoretic speed-up properties such as speedable, effectively speedable, levelable, and effectively levelable introduced by Blum and Marques. In particular, we study the interplay between an r.e. basis A for a subspace V of V ∞ and V with regard to these properties. We show for example that if A or V is speedable , then V is levelable (...) . We also show that if A is an r.e. subset of a recursive basis for V ∞ , then A is levelable iff V is speedable while it is not the case that A is levelable iff V is speedable. We also provide several contrasts between the lattice of r.e. subspaces and the lattice of r.e. sets with respect to these speed-up properties. For example, every maximal set is levelable but we show that there exist supermaximal spaces which are nonspeedable in all possible r.e. degrees as well as supermaximal spaces which are levelable in all r.e. degrees which contain levelable sets. (shrink)

We show that the _sQ_-degree of a hypersimple set includes an infinite collection of \(sQ_1\) -degrees linearly ordered under \(\le _{sQ_1}\) with order type of the integers and each c.e. set in these _sQ_-degrees is a hypersimple set. Also, we prove that there exist two c.e. sets having no least upper bound on the \(sQ_1\) -reducibility ordering. We show that the c.e. \(sQ_1\) -degrees are not dense and if _a_ is a c.e. \(sQ_1\) -degree such that \(o_{sQ_1}, then there exist (...) infinitely many pairwise _sQ_-incomputable c.e. _sQ_-degrees \(\{c_i\}_{i\in \omega }\) such that \((\forall \,i)\;(a. (shrink)

We show: for every noncomputable c.e. incomplete c-degree, there exists a nonspeedable c-degree incomparable with it; The c-degree of a hypersimple set includes an infinite collection of \-degrees linearly ordered under \ with order type of the integers and consisting entirely of hypersimple sets; there exist two c.e. sets having no c.e. least upper bound in the \-reducibility ordering; the c.e. \-degrees are not dense.

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)

We prove that if $A$, $B$ are noncomputable c.e. sets, $A<_{sQ_{1}}B$ and [($B$ is not simple and $A\oplus B\leq _{sQ_{1}}B$) or $B\equiv _{sQ_{1}}B\times \omega $], then there exist infinitely many pairwise $sQ_{1}$-incomparable c.e. sets $\{C_{i}\}_{i\in \omega }$ such that $A<_{sQ_{1}}C_{i}<_{sQ_{1}}B$, for all $i\in \omega $. We also show that there exist infinite collections of $sQ_{1}$-degrees $\{\boldsymbol {a_{i}}\}_{i\in \omega }$ and $\{\boldsymbol {b_{i}}\}_{i\in \omega }$ such that for every $i, j,$ (1) $\boldsymbol {a_{i}}<_{sQ_{1}}\boldsymbol {a_{i+1}}$, $\boldsymbol {b_{j+1}}<_{sQ_{1}}\boldsymbol {b_{j}}$ and $\boldsymbol {a_{i}}<_{sQ_{1}}\boldsymbol {b_{j}}$; (...) (2) every c.e set in $\boldsymbol {a_{i}}$ is a maximal set; and (3) every c.e. set in $\boldsymbol {b_{j}}$ is a non-maximal hyperhypersimple set. We prove that a noncomputable c.e. $sQ$-degree contains either only one or infinitely many c.e. $sQ_{1}$-degrees. The $sQ_{1}$-degrees of c.e. cylinders are dense upper semilattice. (shrink)