The group of fractions of a torsion free lcm monoid is torsion free

An extension of properties of symmetric group to monoids and a pretorsion theory in the category of mappings

Several elementary properties of the symmetric group $S_n$ extend in a nice way to the full transformation monoid $M_n$ of all maps of the set $X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some sense the torsion part of the monoid $M_n$. More precisely, there is a pretorsion theory in the category of all maps $f\colon X\to X$, $X$ an arbitrary finite non-empty set, in which bijections are exactly the torsion objects.

Groupes de Garside

Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups contains the braid groups, all spherical Artin groups, and various generalizations previously considered (In particular, the Garside groups considered in [1,7] are special cases of those considered here; the latter had been called "small Gaussian" in earlier works; the name has been changed in order to uniformize the terminology with works by Bessis, Charney, Mi…

Covers of acts over monoids II

In 1981 Edgar Enochs conjectured that every module has a flat cover and finally proved this in 2001. Since then a great deal of effort has been spent on studying different types of covers, for example injective and torsion free covers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the `other' type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective cove…

torsion群与Bienvenu-Geroldinger猜想

Torsion groups and the Bienvenu--Geroldinger conjecture

本文研究了在H和K为可约monoids的情况下，torsion群的约简有限幂monoid是否同构的问题，并给出了肯定答案。

Monoids of IG-type and Maximal Orders

Let G be a finite group that acts on an abelian monoid A. If f: A -> G is a map so that f(a f(a)(b)) = f(a)f(b), for all a, b in A, then the submonoid S = {(a, f(a)) | a in A} of the associated semidirect product of A and G is said to be a monoid of IG-type. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group Sym_n of degree n, then these monoids first appeared in the work of Gateva-Ivanova and Van den Bergh (they are called monoids of I-type) and later in the work of Jespers and Okninski. It turns out that their associated semigroup algebras s…

关于乘积为1序列幺半群的同构问题

On the isomorphism problem for monoids of product-one sequences

本文研究乘积为1序列幺半群的同构能否决定底层群的同构，并将此前阿贝尔群情形的结果推广到任意挠群。

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang--Baxter equation

Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M…

Abelian quotients of the $Y$-filtration on the homology cylinders via the LMO functor

We construct a series of homomorphisms from the $Y$-filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3\mathcal{IC}_{g,1}/Y_4$ of the $Y$-filtration.

图辫群同调中所有有限阶挠元

Torsion of every finite order in the homology of graph braid groups

本文确定了图辫群同调中所有有限阶挠元的结构，给出了显式代表元、最小图条件及无挠性判定，并证明了奇挠元首次出现的具体位置。

Solutions sets to systems of equations in hyperbolic groups are EDT0L in PSPACE

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in $\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and $\mathsf{NSPACE}(n^4\log n)$ in the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Jeż, Diekert and others on $\mathsf{PSPACE}$ algorithms to solve equations in free mon…

图积群中的二次方程与周期性指数

Quadratic Equations in Graph Products of Groups and the Exponent of Periodicity

本文研究图积群中二次方程的解集周期性指数性质，通过正常形式定义周期性指数，并证明其在图积群下保持不变，适用于右角Artin群及某些无 torsion nilpotent 和 hyperbolic 群。

可容许幺半群作用的 Ore 局部化及其在熵 $-$ 加法公式与桥定理中的应用

Ore localization of amenable monoid actions and applications towards entropy $-$ addition formulas and the bridge theorem

本文构造了可容许幺半群作用的 Ore 局部化与余局部化并证明其保持熵，借此将拓扑熵加法定理与桥定理扩展至任意紧致群，并导出离散 Abel 群上代数熵的加法定理。