Related papers: Some properties of B\"uchi Arithmetics

The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a…

Let $d \ge 2, h \ge 1$ be integers. Using a fragmentation technique, we characterise $(h+1)$-tuples $(R_1, \dots, R_h, R)$ of non-empty families of partitions of $\{1, \dots, d\}$ such that it suffices for an order-$d$ tensor to have…

An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local'…

In the present paper, we consider Presburger arithmetic PrA and the theory of real closed fields RCF. Due to quantifier elimination in these theories, there are two kinds of natural ways to axiomatize them. Namely, on one hand, PrA can be…

We continue the investigation of Boolean-like algebras of dimension n (nBA) having n constants e1,...,en, and an (n+1)-ary operation q (a "generalised if-then-else") that induces a decomposition of the algebra into n factors through the…

The 2-adic valuation of an integer n which is the exponent of the highest power of 2 that divides n. In this paper, we give representations of certain restricted partition functions in terms of 2-adic valuation.

A topological space $X$ is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that whenever the space $X$ has a resolving subset that can be written as an at most countably infinite…

We examine the noncommutative minimal model program for orders on arithmetic surfaces, or equivalently, arithmetic surfaces enriched by a Brauer class $\beta$. When $\beta$ has prime index $p>5$, we show the classical theory extends with…

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by a non-zero element are shown to form an invariant of A, called its double sign. The…

The symbolic representation of a number should be considered as a data structure, and the choice of data structure depends on the arithmetic operations that are to be performed. Numbers are almost universally represented using position…

Bar Codes are combinatorial objects encoding many properties of monomial ideals. In this paper we employ these objects to study Janet-like divisions. Given a finite set of terms U, from its Bar Code we can compute the Janet-like…

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We…

The Feferman-Vaught theorem provides a way of evaluating a first order sentence $\varphi$ on a disjoint union of structures by producing a decomposition of $\varphi$ into sentences which can be evaluated on the individual structures and the…

We prove that linear extensions of the Bruhat order of a matroid are shelling orders and that the barycentric subdivision of a matroid is a Coxeter matroid, viewing barycentric subdivisions as subsets of a parabolic quotient of a symmetric…

We present a labelled sequent calculus for Boolean BI, a classical variant of O'Hearn and Pym's logic of Bunched Implication. The calculus is simple, sound, complete, and enjoys cut-elimination. We show that all the structural rules in our…

We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gr\"obner…

B\"uchi's theorem states that $\omega$-regular languages are characterized as languages of the form $\bigcup_i U_i V_i^\omega$, where $U_i$ and $V_i$ are regular languages. Parikh automata are automata on finite words whose transitions are…

A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_\alpha-x}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study…

Given a suitably nested family $Z = \langle Z(m,k,\gamma) \rangle_{m,k \in \mathbb N, \gamma >0}$ of Borel subsets of matrices, and associated Borel measures and rate function, $\mu$, an entropy, $\chi^{\mu}(Z)$, is introduced which…

We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is…