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The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When…

To an inverse semigroup, we associate an \'etale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that…

Assume that $X$ and $Y$ are arithmetic schemes, i.e., integral schemes of finite types over $Spec(\mathbb{Z})$. Then $X$ is said to be quasi-galois closed over $Y$ if $X$ has a unique conjugate over $Y$ in some certain algebraically closed…

We define the notion of a semicharacter of a group G : A function from the group to C*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is…

Let $G$ be a group. An automorphism of $G$ is called intense if it sends each subgroup of $G$ to a conjugate; the collection of such automorphisms is denoted by $\mathrm{Int}(G)$. In the special case in which $p$ is a prime number and $G$…

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an…

A subsemigroup $S$ of an inverse semigroup $Q$ is a left I-order in $Q$ if every element in $Q$ can be written as $a^{-1}b$ where $a,b \in S$ and $a^{-1}$ is the inverse of $a$ in the sense of inverse semigroup theory. If we insist on $a$…

We show that the topological full group of a Hausdorff ample groupoid with compact unit space coincides with the group of homotopy classes of invertible isometries in pseudofunction algebras associated with the groupoid. Moreover, if the…

Let $G\subset GL_n(k)$ be a finite subgroup and $k[x_1,\dots, x_n]^G\subset k[x_1,\dots, x_n]$ its ring of invariants. We show that, in many cases, the automorphism group of $k[x_1,\dots, x_n]^G$ is $k^\times$. Version 2: Incorporates parts…

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any…

We utilize the structure of quasiautomorphic forms over a Hecke triangle group to define a mapping from a quasiautomorphic form to a vector-valued automorphic form (vvaf). This kind of vvaf we call a Hecke vector-form. First we supply a…

Let G be a group and f be an endomorphism of G. A subgroup H of G is called f-inert if the meet of Hf and H has finite index in the image Hf. The subgroups that are f-inert for all inner automorphisms of G are widely known and studied in…

We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a `non-trivial' example of a pair of $2$-graphs that are not…

A completely inverse $AG^{**}$-groupoid is a groupoid satisfying the identities $(xy)z=(zy)x$, $x(yz)=y(xz)$ and $xx^{-1}=x^{-1}x$, where $x^{-1}$ is a unique inverse of $x$, that is, $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}$. First we…

Let $G$ be an affine algebraic group over an algebraically closed field $k$ of characteristic zero. In this paper, we consider finite $G$-equivariant morphisms $F:X\to Y$ of irreducible affine $G$-varieties. First we determine under which…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

In math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups.…

Let $\mathcal C$ be a subcategory of the category of topologized semigroups and their partial continuous homomorphisms. An object $X$ of the category ${\mathcal C}$ is called ${\mathcal C}$-closed if for each morphism $f:X\to Y$ of the…

In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are…

Given a category, one may construct slices of it. That is, one builds a new category whose objects are the morphisms from the category with a fixed codomain and morphisms certain commutative triangles. If the category is a groupoid, so that…