Related papers: Ideals in $\mathcal{P} _G$ and $\beta G$

Let $G$ be a group, $\beta G$ is the Stone-$\check{C}$ech compactification of $\beta G$ endowed with the structure of a right topological semigroup, $G^*=\beta G\setminus G$. Given any subset $A$ of $G$ and $p\in G^*$, we define the…

Solecki has shown that a broad natural class of $G_{\delta}$ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed…

We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta…

Let $G$ be a group and let $X$ be a transitive $G$-space. We classify the subsets of $X$ with respect to a translation invariant ideal $\mathcal{J}$ in the Boolean algebra of all subsets of $X$, introduce and apply the relative…

We consider the class $\mathcal{A}_0$ of Abelian block-rigid $CRQ$-groups of ring type. A subgroup $A$ of an Abelian group $G$ is called an \textsf{absolute ideal} of the group $G$ if $A$ is an ideal in any ring on $G$. We describe…

A discrete subset $S$ of a topological group $G$ is called a {\it suitable set} for $G$ if $S\cup \{e\}$ is closed in $G$ and the subgroup generated by $S$ is dense in $G$, where $e$ is the identity element of $G$. In this paper, the…

For a discrete group $G$, we consider certain ideals $\mathcal{I}\subset c_0(G)$ of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C$^\ast$-algebra of $G$ and the C$^\ast$-completion…

In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to \'etale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in…

For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the variety of…

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their…

Let $\Gamma$ be a discrete group. To every ideal in $\ell^{\infty}(\G)$ we associate a C$^*$-algebra completion of the group ring that encapsulates the unitary representations with matrix coefficients belonging to the ideal. The general…

For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open…

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional…

For every infinite cardinal $\kappa$ with $\kappa^+=2^\kappa$ we construct a group $G$ of cardinality $|G|=\kappa^+$ such that (i) $G$ is $36$-Shelah, which means that $A^{36}=G$ for any subset $A\subseteq G$ of cardinality $|A|=|G|$; (ii)…

We classify, in terms of the structure of the finite group G, all group algebras KG for which all right ideals are right annihilators of principal left ideals. This means in the language of coding theory that we classify code-checkable…

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property…

We say that an inclusion of an algebra $A$ into a $C^*$-algebra $B$ has the ideal separation property if closed ideals in $B$ can be recovered by their intersection with $A$. Such inclusions have attractive properties from the point of view…

Let $X$ be a completely regular topological space. We assign to each (set theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded complex valued mappings on $X$ equipped with the supremum norm. We…

For a finite group $G$, we associate the quantity $\beta(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second…

Let $G$ be a locally compact group. Then for every $G$-space $X$ the maximal $G$-proximity $\beta_G$ can be characterized by the maximal topological proximity $\beta$ as follows: $$ A \ \overline{\beta_G} \ B \Leftrightarrow \exists V \in…