Related papers: Arithmetic Sets in Groups
It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian…
Arithmetical properties of a finite group are properties of the group which are defined by its arithmetical parameters such as the order of the group, the element orders and so on. In this paper, we discuss a number of results on…
We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in…
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…
The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.
We entirely classify definable sets up to definable bijections in $\mathbb{Z}$-groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable…
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by…
A notion of arithmetic similarity between number fields is defined by requiring equality of some arithmetic statistics over all but finitely many rational primes. The exceptional set is empty in all previously studied cases, but existing…
We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a…
We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear…
A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more…
We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here,…
We define an algebraic group over a group $G$ to be a variety - that is, a subset of $G^d$ defined by equations over $G$ - endowed with a group law whose coordinates can be expressed as word maps. In the case where $G$ is a torsion-free…
In this paper we highlight a few open problems concerning maximal sum-free sets in abelian groups. In addition, for most even order abelian groups $G$ we asymptotically determine the number of maximal distinct sum-free subsets in $G$. Our…
An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group.…
We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
We study the set of Dedekind cuts over a linearly ordered Abelian group as a structure over the language (0,<,+,-). Moreover, we obtain a simple set of axioms for the universal part of the theory of such structures. Finally, we prove that…