Related papers: A characterization of strongly dependent ordered A…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

In \cite{HPP}, Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that $T$ has the canonical base property in a strong form, " internality to" being replaced by…

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is…

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of…

We prove that for any $\alpha\in[0,\omega_1)$ there exists a strongly separately continuous function $f:\ell_\infty\to [0,1]$ such that $f$ belongs to the $(\alpha+1)$'th /$(\alpha+2)$'th/ Baire class and does not belong to the $\alpha$'th…

The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n=1. In this paper, we consider two families of…

Elder, Kambites, and Ostheimer showed that if the word problem of a finitely generated group $H$ is accepted by a $G$-automaton for an abelian group $G$, then $H$ is virtually abelian. We give a new, elementary, and purely combinatorial…

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only…

We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ \alpha\in\omega+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $…

We give an example of a finite rank, in fact aleph-1 categorical theory where the CBP (canonical base property) does not hold. We include a "group-like" example. We also prove, in a finite Morley rank context, if all definable Galois groups…

In this paper we define Ordered Generating System for finite non-abelian groups, which is a generalization of the basis theorem for finite abelian groups. We prove the following: If each composition factor of a group G has Ordered…

In this note we give a classification of the Maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analogue of Cartan's theorem that all maximal tori in a connected compact Lie group are conjugate.

\'{E}tale difference algebraic groups are a difference analog of \'{e}tale algebraic groups. Our main result is a Jordan-H\"{o}lder type decomposition theorem for these groups. Roughly speaking, it shows that any \'{e}tale difference…

It is a theorem of Artin, Tits et al. that a finite simple group is determined by its order, with the exception of the groups (A_3(2), A_2(4)) and (B_n(q), C_n(q)) for n > 2, q odd. We investigate the situation for finite semisimple groups…

Let $K/F$ be a finite Galois extension of number fields with Galois group $G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/ K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich groups of…

We adapt the theory of partition rank and analytic rank to the category of abelian groups. If $A_1, \dots, A_k$ are finite abelian groups and $\phi : A_1 \times \cdots \times A_k \to \mathbf{T}$ is a multilinear map, where $\mathbf{T} =…

We discuss the notion of strongly unbounded type for abelian length categories; this is closely related to the Second Brauer-Thrall Conjecture for artin algebras. A new ingredient is the space of characters in the sense of Crawley-Boevey.

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio $r\geq 2$. This extends the case $r=1$ studied in previous papers \cite{1,8,4}.

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$),…

In mathematical physics the Galois top, introduced by S. Adlaj, possesses a fixed point on one of two Galois axes through its center of mass. This heavy top has two algebraic motion invariants and an additional transcendental…