Related papers: The first-order theory of $\ell$-permutation group…
It is shown that $2\beta_1(\G)\leq h(\G)$ for any countable group $\G$, where $\beta_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first…
We present a higher well-ordering principle which is equivalent (over Simpson's set theoretic version of $\text{ATR}_0$) to the existence of transitive models of Kripke-Platek set theory, and thus to $\Pi^1_1$-comprehension. This is a…
We consider the translational hull $\Omega(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by…
Let $\Omega$ be a set equipped with an equivalence relation $\sim$; we refer to the equivalence classes as blocks of $\Omega$. A permutation group $G \le \mathrm{Sym}(\Omega)$ is $k$-by-block-transitive if $\sim$ is $G$-invariant, with at…
A classical result of Sherman says that if the space of self-adjoint elements in a $C^*$-algebra $\mathcal{A}$ is a lattice with respect to its canonical order, then $\mathcal{A}$ is commutative. We give a new proof of this theorem which…
Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are: (a) being atomic, i.e. not being…
We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a…
An abelian lattice-ordered group, or abelian $\ell$-group, is an abelian group equipped with a compatible lattice ordering. In this paper, we introduce two multi-sorted extensions of abelian lattice-ordered groups inspired by the zero-set…
We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the $\Sigma_1$ theory is undecidable (already over two letters). We investigate the decidability…
We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $\Delta^1_2$ linear order similar to an antichain in $2^{<\omega_1}$ ordered lexicographically or a certain Borel partial order $\leq_0$…
Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain {a: M models P(a)}. Now suppose T is a complete first order theory in L with…
We show that several classes of ordered structures (namely, convex linear orders, layered permutations, and compositions) admit first-order logical limit laws.
We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$…
We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the…
In this sequel paper, we continue the analysis of the prime order element graph $\Gamma(G)$ of a finite group $G$, where vertices are elements of $G$ and edges connect distinct elements $x, y$ satisfying $\circ(xy) = p$ for some prime $p$.…
The paper is a first of two and aims to show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic…
For a graph with edge ordering, a linear order on the edge set, we obtain a permutation of vertices by considering the edges as transpositions of endvertices. It is known from D\'enes' results that the permutation of a tree is a full cyclic…
The notion of $n$-transitivity can be carried over from groups of diffeomorphisms on a manifold $M$ to groups of bisections of a Lie groupoid over $M$. The main theorem states that the $n$-transitivity is fulfilled for all $n\in\mathbb N$…
Originally introduced by Kolmann and Shelah as a surrogate for saturated models, limit models have been established as natural and useful objects when studying abstract elementary classes. Shelah began the study of when (multiple notions…
Let x be an element of a group G. For a positive integer n let E_n(x) be the subgroup generated by all commutators [...[[y,x],x],...,x] over y in G, where x is repeated n times. There are several recent results showing that certain…