伪有限集，伪o-极小性

Sharp o-极小性与格点计数

Sharp o-minimality and lattice point counting

本文在sharp o-极小结构与$\mathbb{R}_{\exp}$的语境下，分别给出了Barroero和Widmer关于o-极小可定义族纤维格点计数精确估计结果的有效版本。

Pseudo-jump inversion and SJT-hard sets

There are noncomputable c.e.\ sets, computable from every SJT-hard c.e.\ set. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone.

Cell decomposition and classification of definable sets in p-optimal fields

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-optimal fields satisfying the Extreme Value Property (a property which in particular holds in fields which are elementarily equivalent to a p-adic one). For such fields K, we prove that every definable subset of KxK^d whose fibers are inverse images by the valuation o…

Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets

We introduce Kleene-Varlet spaces as partially ordered sets equipped with a polarity satisfying certain additional conditions. By applying Kleene-Varlet spaces, we prove that each regular pseudocomplemented Kleene algebra is isomorphic to a subalgebra of the rough set regular pseudocomplemented Kleene algebra defined by a tolerance induced by an irredundant covering. We also characterize the Kleene-Varlet spaces corresponding to the regular pseudocomplemented Kleene algebras satisfying the Stone identity.

严格o-极小结构与严格单元分解

Sharply o-minimal structures and sharp cellular decomposition

本文研究严格o-极小结构，提出其理论并证明了通过约简可实现严格单元分解，从而控制可定义集的几何复杂性。

Tame topology over definable uniform structures

A visceral structure on M is given by a definable base for a uniform topology on its universe in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. This context includes o-minimal ordered groups, p-adic fields, and other examples. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable unary function on M has a finite set of discontinuities; any definable function on a Cartesian power of M is continuous on a nonempty open set; and assuming definable fi…

Model completeness of o-minimal fields with convex valuations

We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definabl…

Remarks on small sets related to trigonometric series

We show that several classes of sets, like N_0-sets, Arbault sets, N-sets and pseudo-Dirichlet sets are closed under adding sets of small size.

Model-completeness for the lattice of finite unions of closed intervals of a dense linear order

Let I be a dense linear order with a left endpoint but no right endpoint. We consider the lattice L(I) of finite unions of closed intervals of I. This lattice arises naturally in the setting of o-minimality, as these are precisely the closed definable sets in any o-minimal expansion of I. Our main result says that L(I), the expansion of the lattice by constants for the empty set and the smallest element of I (viewed as a singleton subset) as well as four unary functions, is model-complete. The proof of the main result makes use of previous results regarding the weak monadic second order theory…