Related papers: A Cauchy-Davenport theorem for semigroups

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector…

Cayley's theorem tells us that all groups $\mathbf{G}$ occur as subgroups of the group of automorphisms over some set $X$. In this paper we consider a `sort-of' converse to this question: given a set $X$ and some transformation group…

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1,…

In this paper, we study definably compact semigroups in o-minimal structures, aiming to extend the theory of definable groups to a broader algebraic setting. We show that any definably compact semigroup contains idempotents and admits a…

We are interested in semigroups of the form $\langle G,a\rangle\setminus G$, where $G$ is a permutation group of degree $n$ and $a$ a non-permutation on the domain of $G$. A theorem of the first author, Mitchell and Schneider shows that, if…

Given a $\Gamma$-semigroup $S$, we construct a semigroup $\Sigma$ in such a way that one sided ideals and quasi-ideals of $S$ can be regarded as one sided ideals and quasi-ideals respectively of $\Sigma$. This correspondence and other…

Let $T(X)$ (resp. L(V)) be the semigroup of all transformations (resp. linear transformations) of a set $X$ (resp. vector space $V$). For a subset $Y$ of $X$ and a subsemigroup $\mathbb{S}(Y)$ of $T(Y)$, consider the subsemigroup…

Suppose that $X$ be a nonempty set. Denote by $\mathcal{T}(X)$ the full transformation semigroup on $X$. For $\varnothing \neq Z\subseteq Y\subseteq X$, let $\mathcal{T}(X,Y,Z)=\{\alpha \in \mathcal{T}(X): Y\alpha \subseteq Z \}$. Then…

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered string of terms from $G$ with repetition allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity…

We generalize the small cancellation theory over hyperbolic groups developed by Olshanskii to the case of relatively hyperbolic groups. This allows us to construct infinite finitely generated groups with exactly $n$ conjugacy classes for…

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be…

Let $G$ be a finite group and let $k \geq 2$. We prove that the coprime subgroup $\gamma_k^*(G)$ is nilpotent if and only if $|xy|=|x||y|$ for any $\gamma_k^*$-commutators $x,y \in G$ of coprime orders (Theorem A). Moreover, we show that…

Let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the…

We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is…

Let $G$ be a finite abelian group of exponent $n$ and let $A$ be a non-empty subset of $[1,n-1]$. The Davenport constant of $G$ with weight $A$, denoted by $D_A(G)$, is defined to be the least positive integer $\ell$ such that any sequence…

Let $X$, $Y$ be closed irreducible subvarieties of an absolutely simple abelian variety of dimension $g$ over a field. If $\dim(X) + \dim(Y) \le g$, we prove that the addition morphism $X \times Y \to X + Y$ is semismall. As a consequence,…

Let $p$ a large enough prime number. When $A$ is a subset of $\mathbb{F}_p\smallsetminus\{0\}$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\mathbb{F}_p\smallsetminus\{0\}\subset A(A+A)$. In this…

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups…

A smallest generating set of a semigroup is a generating set of the smallest cardinality. Similarly, an irredundant generating set $X$ is a generating set such that no proper subset of $X$ is also a generating set. A semigroup $S$ is…

For semisimple groups, possibly multiplied by U(1)'s, the number of Yang-Mills gauge fields is equal to the number of generators of the group. In this paper, it is shown that, for non-semisimple groups, the number of Yang-Mills fields can…