A valued field is said to have the definable-absolute-valuational-closure property if, in the language of valued fields, every definable set in the first-order structure of said valued field is dense, with respect to an induced valuation, inside a definable set in the first-order structure of the algebraic closure of said field, both with parameters in the smaller field. Every omega-free pseudo-algebraically closed non-trivially valued field with a finite exponent of imperfection is proved to have this property.
This is the first of two papers on the fine structure of HOD in models of the Axiom of Determinacy ([Formula: see text]). Let [Formula: see text]. [12] shows that under a natural hypothesis on the existence of iteration strategies, the basic fine structure theory for pure extender models goes over to [Formula: see text]. In this paper, we prove a fine condensation theorem, quite similar to Theorem 9.3.2 of Zeman’s book [18], except that condensation for iteration strategies has been added to the mix. In the second paper [15], we use this theorem to analyze and prove basic facts about the Dodd parameters for lbr hod mice. The results of these two papers will be used to show that in [Formula: see text], [Formula: see text] holds iff [Formula: see text] is not subcompact.
Suppose [Formula: see text] is elementarily equivalent to an archimedean ordered abelian group [Formula: see text] with small quotients (for all [Formula: see text], [Formula: see text] is finite). Then every stable reduct of [Formula: see text] which expands [Formula: see text] (equivalently every reduct that does not add new unary definable sets) is interdefinable with [Formula: see text]. This extends previous results on stable reducts of [Formula: see text] to (stable) reducts of elementary extensions of [Formula: see text]. In particular this holds for [Formula: see text] and [Formula: see text]. As a result, we answer a question of Conant from 2018. This result is a corollary of a more general statement about expansions of weakly-minimal 1-based expansions of abelian groups with small quotients preserving the [Formula: see text] operator.
We answer the question: which levels of Selivanov's fine hierarchy contain new Turing degrees?
In this paper, we show that every deconstructible class of modules with all embeddings, all pure embeddings and all RD-embeddings is stable. The argument is presented in the context of abstract classes of modules without amalgamation and the key idea is to construct a stable-like independence relation. In particular, the following classes of modules with all embeddings, all pure embeddings and all RD-embeddings are shown to be stable: all free and torsion-free modules over any ring, and for each n >= 0, the classes of all modules of projective and flat dimension <= n over any ring, and the class of all modules of injective dimension <= n over any right noetherian ring.
