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Register a new userCoherent presentations of a class of monoids admitting a Garside family
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This paper shows how to construct coherent presentations of a class of monoids, including left-cancellative noetherian monoids containing no nontrivial invertible element and admitting a Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of Delignes original construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to a dual braid monoid, and to some monoids which are neither Artin-Tits nor Garside. For the Artin-Tits monoid of type $widetilde{A}_{2}$, a finite coherent presentation is given, having a finite Garside family as a generating set.
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We compute coherent presentations of Artin monoids, that is presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squiers and Knuth-Bendixs completions into a homotopical completion-reduction, applied to Artins and Garsides presentations. The main result of the paper states that the so-called Tits-Zamolodchikov 3-cells extend Artins presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.
Let $G$ be a finite group admitting a coprime automorphism $phi$ of order $n$. Denote by $G_{phi}$ the centralizer of $phi$ in $G$ and by $G_{-phi}$ the set ${ x^{-1}x^{phi}; xin G}$. We prove the following results. 1. If every element from $G_{phi}cup G_{-phi}$ is contained in a $phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_{phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x in G_{-phi}$ and any two elements of $G_{-phi}$ are contained in a $phi$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,phi]$ is bounded in terms of $c,d,e,n$.
A variety is finitely based if it has a finite basis of identities. A minimal non-finitely based variety is called limit. A monoid is aperiodic if all its subgoups are trivial. Limit varieties of aperiodic monoids have been studied by Jackson, Lee, Zhang and Luo, Gusev and Sapir. In particular, Gusev and Sapir have recently reduced the problem of classifying all limit varieties of aperiodic monoids to the two tasks. One of them is to classify limit varieties of monoids satisfying $xsxt approx xsxtx$. In this paper, we completely solve this task. In particular, we exhibit the first example of a limit variety of monoids with countably infinitely many subvarieties. In view of the result by Jackson and Lee, the smallest known monoid generating a variety with continuum many subvarieties is of order six. It follows from the result by Edmunds et al. that if there exists a smaller example, then up to isomorphism and anti-isomorphism, it must be a unique monoid $P_2^1$ of order five. Our main result implies that the variety generated by $P_2^1$ contains only finitely based subvarieties and so has only countably many of them.
A limit variety is a variety that is minimal with respect to being non-finitely based. The two limit varieties of Marcel Jackson are the only known examples of limit varieties of aperiodic monoids. Our previous work had shown that there exists a limit subvariety of aperiodic monoids that is different from Marcel Jacksons limit varieties. In this paper, we introduce a new limit variety of aperiodic monoids.
This is a survey of results on random group presentations, and on random subgroups of certain fixed groups. Being a survey, this paper does not contain new results, but it offers a synthetic view of a part of this very active field of research.
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