arXiv subjects

Ambroise Grau

Publications and source records attributed to Ambroise Grau.

4 recordsLinked to original sources

arXiv2026

The endomorphism tower of a finite symmetric group

We consider the endomorphism tower of a monoid $M$, that is, the sequence of monoids End$_i(M)$ where End$_0(M)=M$ and for all $i\geq 1$, End$_i(M)$ is the monoid of all endomorphisms of End$_{i-1}(M)$. We show that for a finite monoid $M$ this sequence does not stabilise in a finite number of steps. Our focus is then on the case where $M=\mathcal{S}_n$, the symmetric group on a finite number $n$ of points. It is well known that other than in exceptional cases (which are avoided by taking $n \geq 7$), the corresponding automorphism tower of $\mathcal{S}_n$ stabilises at the first step. In spite of the natural nature of this question, nothing was known of the endomorphism tower above the level $i=1$. We determine (for each $n \geq 7)$ the elements of End$_2(\mathcal{S}_n)$ and their multiplication and thus verify that the monoids End$_i(\mathcal{S}_n)$ for $i=0,1,2$ all have group of units isomorphic to $\mathcal{S}_n$. We show that the same is true of End$_3(\mathcal{S}_n)$.

math.GR↗

arXiv2024

Translational hulls of semigroups of endomorphisms of an algebra

We consider the translational hull $Ω(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by transformations, or by endomorphisms, of $A$. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of $I$ and the idealiser of $I$ within $\mathrm{End}(A)$, which in the case where $I$ is an ideal is simply $\mathrm{End}(A)$. We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning $Ω(I)$ from the translational hull $Ω(I/{\approx})$ of a quotient $I/{\approx}$ of $I$. We illustrate these concepts in detail in the cases where $A$ is: a free algebra; an independence algebra; a finite symmetric group.

math.RA↗

arXiv2023

The structure of End($\mathcal{T}_n$)

The full transformation semigroups $\mathcal{T}_n$, where $n\in \mathbb{N}$, consisting of all maps from a set of cardinality $n$ to itself, are arguably the most important family of finite semigroups. This article investigates the endomorphism monoid End($\mathcal{T}_n$) of $\mathcal{T}_n$. The determination of the elements of End($\mathcal{T}_n$) is due Schein and Teclezghi. Surprisingly, the algebraic structure of End($\mathcal{T}_n$) has not been further explored. We describe Green's relations and extended Green's relations on End($\mathcal{T}_n$), and the generalised regularity properties of these monoids. In particular, we prove that $\mathcal{H}=\mathcal{L} \subseteq \mathcal{R}= \mathcal{D}=\mathcal{J}$ (with equality if and only if $n=1$); the idempotents of End($\mathcal{T}_n$) form a band (which is equal to End($\mathcal{T}_n$) if and only if $n=1$) and also the regular elements of End($\mathcal{T}_n$) form a subsemigroup (which is equal to End($\mathcal{T}_n$) if and only if $n\leq 2$). Further, the regular elements of End($\mathcal{T}_n$) are precisely the idempotents together with all endomorphisms of rank greater than $3$. We also provide a presentation for End($\mathcal{T}_n$) with respect to a minimal generating set.

math.RA↗

arXiv2022

The semigroup of endomorphisms with restricted range of an independence algebra

Since its introduction by Symons, the semigroup of maps with restricted range has been studied in the context of transformations on a set, or of linear maps on a vector space. Sets and vector spaces being particular examples of independence algebras, a natural question that arises is whether by taking the semigroup $T(\mathcal{A},\mathcal{B})$ of all endomorphisms of an independence algebra $\mathcal{A}$ whose image lie in a subalgebra $\mathcal{B}$, one can obtain corresponding results as in the cases of sets and vector spaces. In this paper, we put under a common framework the research from Sanwong, Sommanee, Sullivan, Mendes-Gonçalves and all their predecessors. We describe Green's relations as well as the ideals of $T(\mathcal{A},\mathcal{B})$ following their lead. We then take a new direction, completely describing all of the extended Green's relations on $T(\mathcal{A},\mathcal{B})$. We make no restriction on the dimension of our algebras as the results in the finite and infinite dimensional cases generally take the same form.

math.RA↗