Braids and partial permutations

We study the inverse braid monoid IBn introduced by Easdown and Lavers in 2004. We completely describe the factorizable structure of IBn and use this to give a new proof of the Easdown-Lavers presentation; we also derive several new presentations, each of which gives rise to a new presentation of the symmetric inverse monoid. We then define and study the pure inverse braid monoid IPn which is related to IBn in the same way that the pure braid group is related to the braid group.     

The inverse braid monoid

We introduce an inverse monoid which plays a similar role with respect to the symmetric inverse semigroup that the braid group plays with respect to the symmetric group.     

Multifraction reduction IV: Padding and Artin–Tits monoids of sufficiently large type

We investigate the padded version of reduction, an extension of multifraction reduction as defined in arXiv:1606.08991, and connect it both with ordinary reduction and with the so-called Property H. As an application, we show that all Artin–Tits groups of sufficiently large type satisfy some weakened version Conjecture A^padded of Conjecture A, thus showing that the reduction approach is relevant for these groups.     

Self-adjunctions and matrices

It is shown that the multiplicative monoids of Temperley-Lieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. This self-adjunction underlies the orthogonal group case of Brauer's representation of the Brauer centralizer algebras.     

A finitely presented monoid with a non-finitely generated group of units

We exhibit an example of a monoid defined by finitely many generators and defining relations, the group of units of which is not finitely generated. Received: 30 August 2006 Revised: 23 October 2006     

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

Automorphisms of groups

The survey presents classical assertions due to Nielsen, Whitehead, and others, well-known theorems on automorphisms included in monographs on group theory, and recent results in this area. Attention is focused on the progress in automorphism groups theory for free, solvable, modular, and profinite groups. New tools of investigation using graphs and geometrical ideas are also discussed.     

Non-cancellation for certain classes of groups

For a certain class of groups, which are semidirect products arising from an action of a finite rank free abelian group on another group, we study cancellation of the infinite cyclic group in isomorphic direct products. As an application we obtain a sufficient condition for triviality of the genus of certain nilpotent groups.     

On loxodromic actions of Artin–Tits groups

Artin–Tits groups act on a certain delta-hyperbolic complex, called the “additional length complex”. For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that “most” elements of Artin–Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin–Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin–Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.     

WHEN ARE DIVISIBLE ABELIAN GROUPS INJECTIVE?

Abstract

The above question is considered, in the categories ShL of sheaves on a local lattice L and MEns of sets acted upon by a monoid M, for either all divisible abelian groups or all torsionfree divisible abelian groups, the aim being to characterize those L and M for which these types of abelian groups are Injective. Typical results: All divisible abelian groups are injective (i) in ShL iff L is Boolean, (ii) in MEns, M left or right cancellative, iff M is trivial, and (iii) in MEns, M commutative iff M is finite and idempotent.     

The category of 3-computads is not cartesian closed

We show, using [A. Carboni, P.T. Johnstone, Connected limits, familial representability and Artin glueing, Math. Structures Comput. Sci. 5 (1995) 441-459] and Eckmann-Hilton argument, that the category of 3-computads is not cartesian closed. As a corollary we get that neither the category of all computads nor the category of n-computads, for n>2, do form locally cartesian closed categories, and hence elementary toposes.     

On quasi divisor theories and systems of valuations

We study the relationship between divisor theories and systems of valuations, and characterize monoids with quasi divisor theories of finite character by systems of essential valuations. Throughout, we avoid ideal theory but use divisor theoretical methods.     

The group of parenthesized braids

We investigate a group B_• that includes Artin's braid group B_∞ and Thompson's group F. The elements of B_• are represented by braids diagrams in which the distances between the strands are not uniform and, besides the usual crossing generators, new rescaling operators shrink or stretch the distances between the strands. We prove that B_• is a group of fractions, that it is orderable, admits a nontrivial self-distributive structure, i.e., one involving the law x(yz)=(xy)(xz), it embeds in the mapping class group of a sphere with a Cantor set of punctures, and that Artin's representation of B_∞ into the automorphisms of a free group extends to B_•.     

A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities

From Wajnryb's presentation, we extract a simple presentation of the mapping class group of the genus g surface as a quotient of an Artin group by simple relations among the centers of sub-Artin groups. Topological meanings are given by using deformation of simple singularities. Received: 22 January 1998 / in final form: 16 February 1999     

Quantaloids and non-commutative ring representations

Our aim is to generalize to the non-commutative case, the generic representation of commutative rings by sheaves on their quantales of ideals. As the quantale of two-sided ideals is not a sufficiently rich structure, we define and work on a quantaloid of left and right ideals. A workable notion of sheaf is introduced using matrices with values in a quantaloid. For a given ringR, we obtain a category of sheaves where the terminal object is endowed with a special subobject. There exists a representing sheaf forR in the sense that the elements ofR correspond to the sections from the special subobject and the global sections correspond to the center.     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

Monoid generalizations of the Richard Thompson groups

The groups G_k,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_k,1, and to inverse monoids, called ; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_k,1 have connections with circuit complexity (studied in other papers). Here we prove that M_k,1 and are congruence-simple for all k. Their Green relations J and D are characterized: M_k,1 and are J-0-simple, and they have k−1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra O_k. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets.