We investigate issues surrounding an old question of Yates’ as to the existence of a minimal degree with no strong minimal cover, specifically with respect to the hyperimmune-free degrees.

We consider the computably enumerable sets under the relation of Qreducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, 〈RQ,≤Q 〉, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of 〈RQ,≤Q 〉 is undecidable.

Let A be a finitely generated domain of GK dimension less than 3 over a field K and let Q(A) denote the quotient division algebra of A. Using the ideas of Smoktunowicz, we show that if D is a finitely generated division subalgebras of Q(A) of GK dimension at least 2, then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dime...

Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand– Kirillov di...

We investigate and extend the notion of a good approximation with respect to the enumeration (De) and singleton (Ds) degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings ιs and ι̂s of, respectivel...

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.

Ambos-Spies and Kučera [1, Problem 4.5] asked if there is a non-computable set A which is low for the computably random reals. We show that no such A is of hyper-immune degree. Thus, each g ≤T A is dominated by a computable function. Ambos-Spies and Kučera [1, Problem 4.8] also asked if every S-low set is S0-low. We give a partial solution to this problem, showing that no S-low set is of hyper-...

Deformations of maximal supergravity theories induced by gauging non-abelian subgroups of the duality group reveal the presence of charged M-theory degrees of freedom that are not necessarily contained in supergravity. The relation with Mtheory degrees of freedom is confirmed by the representation assignments under the duality group of the gauge charges and the ensuing vector and tensor gauge f...