Related papers: On Infinitary Rational Relations and Borel Sets

The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof…

We define a new class of infinitary logics $\mathscr L^1_{\kappa,\alpha}$ generalizing Shelah's logic $\mathbb L^1_\kappa$ defined in \cite{MR2869022}. If $\kappa=\beth_\kappa$ and $\alpha <\kappa$ is infinite then our logic coincides with…

We study the Borel and analytic subsets of the spaces \({}^{\kappa}\kappa\) and \({}^{\kappa}2\) endowed with ideal topologies, where \(\kappa\) is a regular uncountable cardinal. We establish that the Borel hierarchy does not collapse in…

We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.

Iteration semirings are Conway semirings satisfying Conway's group identities. We show that the semirings $\N^{\rat}\llangle \Sigma^* \rrangle$ of rational power series with coefficients in the semiring $\N$ of natural numbers are the free…

Boris Shapiro and Michael Shapiro have a conjecture concerning the Schubert calculus and real enumerative geometry and which would give infinitely many families of zero-dimensional systems of real polynomials (including families of…

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use $SO(3)$ gauge theory to provide a general criterion…

We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also…

We present a selection of basic results on Borel reducibility of Borel ideals and equivalence relations, especially those with comparably short proofs. The focal point are reducibility/irreducibility results related to some special…

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with…

In this article we provide effective characterisations of regular languages of infinite trees that belong to the low levels of the Wadge hierarchy. More precisely we prove decidability for each of the finite levels of the hierarchy; for the…

This expanded version corrects some misprints of the first version, details completely the poof of Borel-Leroy summability and for $k=3$ in the complex case provides a new improved representation which relies on ordinary convergent Gaussian…

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply…

We prove that all algebraic bases $\beta$ allow an eventually periodic representations of the elements of $\mathbb Q(\beta)$ with a finite alphabet of digits $\mathcal A$. Moreover, the classification of bases allowing that those…

We analyse the fusion of representations of the triplet algebra, the maximally extended symmetry algebra of the Virasoro algebra at c=-2. It is shown that there exists a finite number of representations which are closed under fusion. These…

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^\alpha 2,<_{\text{lex}}\rangle$ for $\alpha$ an ordinal, generalising the setting of the real line, working throughout…

We show that normality for continued fractions expansions and normality for base-$b$ expansions are maximally logically separate. In particular, the set of numbers that are normal with respect to the continued fraction expansion but not…

In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second…

Let $\sigma_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots <k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang…

We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no…