We propose two theories, one generalizing the notion of regularity, the other symmetric to it. Under two additional axioms one obtains model completeness of both theories. Models of these theories can be viewed as rings of sections of sheaves whose stalks are valuation rings. Regular rings correspond to the special case where all stalks are trivial valuation rings, that is fields.

The archetypal Rumely domain is the ring \widetildeZ of algebraic integers. Its constructible Boolean algebra is atomless. We study here the opposite situation: Rumely domains whose constructible Boolean algebra is atomic. Recursive models (which are rings of algebraic numbers) are proposed; effective model-completeness and decidability of the corresponding theory are proved.

We continue the study of a theory which is a valued analogue of the theory of regular rings studied by Carson, Lipshitz and Saracino, characterize it as the model companion of the theory of Prüfer rings, and prove its decidability. We then link it to the theory of p.p. rings developed by Weispfenning and show that it admits quantifier elimination in a related language.

Extending the language of rings to include predicates for Jacobson radical relations, we show that the theory of regular rings defined by Carson, Lipshitz and Saracino is the model completion of the theory of semisimple rings. Removing the requirement on the Jacobson radical (reduced to {0}), we prove that the theory of rings with no nilpotents does not admit a model companion relative to this augmented language.

Developing an analogue of Solovay reducibility in the higher recursion setting, we show that results from the classical computably enumerable case can be extended to the new context.

Sureson, C., Symmetric submodels of a Cohen generic extension, Annals of Pure and Applied Logic 58 247–261. We study some symmetric submodels of a Cohen generic extension and the satisfaction of several properties ) which strongly violate the axiom of choice.

This paper continues the study of covering properties of models closed under countable sequences. In a previous article we focused on C. Chang's Model . Our purpose is now to deal with the model N = ∪ { L [A]: A countable ⊂ Ord}. We study here relations between covering properties, satisfaction of ZF by N , and cardinality of power sets. Under large cardinal assumptions N is strictly included in Chang's Model C , it may thus be interesting to (...) obtain an analogue for N of the covering result for C . As in [13], we say that N satisfies the covering property if, for any set of ordinals X in the Universe, there exists a set Y in N such that X ⊂ Y and |Y| = |X| ℵ0 . In the first part, we show that if N does not satisfy the covering property, then for any α ω 1 , there exists an inner model with α measurable cardinals. Throughout this paper, we compare three situations: 1. V is the universe constructible from a countable sequence of ultrafilters on different measurable cardinals. 2. N does not satisfy the covering property. 3. There exist ω 1 measurable cardinals or a compact cardinal. Section 2 is devoted to the relation between satisfaction of ZF by N and the covering property. If holds, then N satisfies both ZF and the covering property. On the contrary, in case , N satisfies neither ZF nor the covering property. It is thus natural to obtain that the satisfaction of ZF by N implies the satisfaction of the covering property. Finally we study the cardinality of P ∩ N for a cardinal λ in the different cases. For example we show that if N does not satisfy the covering property, then + is inaccessible in any L [ A ] where A is a countable set of ordinals. This is not the case if holds. Let the expression “ λ + is inaccessible in N ” mean that for any α λ + | P ∩ N | ⩽ λ . If situation occurs, then there exists a measurable cardinal κ such that “ κ + is inaccessible in N ”. The large cardinal assumption is necessary since we show that the last fact implies the violation of the covering property by N. (shrink)