Related papers: NIP formulas and Baire 1 definability

We are going to prove that if the theory of a structure $\mathcal M=\langle \mathbb{N}, \Sigma \rangle$ is decidable and the standard order $<$ on natural numbers $\mathbb{N}$ is definable in $\mathcal M$, then there is a nontrivial…

A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable (and the family of simple) first order theories. We examine…

The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component…

Let $A$ be a separable amenable $C^*$-algebra and $B$ a non-unital and $\sigma$-unital simple $C^*$-algebra with continuous scale ($B$ need not be stable). We classify, up to unitary equivalence, all essential extensions of the form $0…

We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and…

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in…

If an automorphism f of a structure M is such that fix(f^k) = fix(f) for all positive k, then M|fix(f) is a substructure of M. The possible isomorphism types of such M|fix(f) are characterized when M is countable and arithmetically…

A partial description of the structure of positive unital maps $\phi: M_2(\bC) \to M_{n+1}(\bC)$ ($n\geq 2$) is given.

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are \Sigma^1_1-complete, in particular the isomorphism relation of dense…

I shall describe a general model-theoretic task to construct expansions of pseudofinite structures and discuss several examples of particular relevance to computational complexity. Then I will present one specific situation where finding a…

We characterize extensions of commutative rings $R \subseteq S$ whose sets of subextensions $[R,S]$ are finite ({\it i.e.} $R\subseteq S$ has the FIP property) and are Boolean lattices, that we call Boolean FIP extensions. Some…

A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{\varphi_m\}_{m=1}^{\infty}$ of real-valued functions $\varphi_m \in~C({\mathbb R}^n)$ and a logarithmically…

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems in the theory of functional spaces is the…

Let $K$ be an NIP field and let $v$ be a henselian valuation on $K$. We ask whether $(K,v)$ is NIP as a valued field. By a result of Shelah, we know that if $v$ is externally definable, then $(K,v)$ is NIP. Using the definability of the…

$(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide…

A qualitative representation $\phi$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^\phi = a^\phi | b^\phi$, as we do for ordinary representations, we only require that $c^\phi\supseteq a^\phi |…

This paper is concerned with the question of when a theory is refutable with certainty on the basis of sequence of primitive observations. Beginning with the simple definition of falsifiability as the ability to be refuted by some finite…

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

An A-infinity bialgebra of type (m,n) is a Hopf algebra H equipped with a "compatible" operation \omega : H^{\otimes m} \to H^{\otimes n} of positive degree. We determine the structure relations for A-infinity bialgebras of type (m,n) and…

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various…