BRAIDS AND ORDER-PRESERVING PARTIAL PERMUTATIONS

We initiate a new way to study presentations of the inverse braid monoid [Formula: see text]. Our method makes use of the monoid [Formula: see text] of all order-preserving partial permutations on an n-element chain. As an application we derive the Easdown–Lavers (2004) presentation of [Formula: see text], and we also obtain new presentations for the symmetric inverse monoid [Formula: see text] and the monoid [Formula: see text] of all order-preserving partial braids with at most n strings.

A PRESENTATION OF THE DUAL SYMMETRIC INVERSE MONOID

The dual symmetric inverse monoid [Formula: see text] is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of [Formula: see text] and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

LARGEST 2-GENERATED SUBSEMIGROUPS OF THE SYMMETRIC INVERSE SEMIGROUP

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.

ON A NEW APPROACH TO THE DUAL SYMMETRIC INVERSE MONOID $\mathcal{I}_{X}^{\ast}$

We construct the inverse partition semigroup[Formula: see text], isomorphic to the dual symmetric inverse monoid[Formula: see text], introduced in [6]. We give a convenient geometric illustration for elements of [Formula: see text]. We describe all maximal subsemigroups of [Formula: see text] and find a generating set for [Formula: see text] when X is finite. We prove that all the automorphisms of [Formula: see text] are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets X, we establish that, up to equivalence, there is a unique faithful effective transitive representation of [Formula: see text], namely to [Formula: see text]. Finally, we construct an interesting [Formula: see text]-cross-section of [Formula: see text], which is reminiscent of [Formula: see text], the [Formula: see text]-cross-section of [Formula: see text], constructed in [4].

Relative Abelian kernels of some classes of transformation monoids

We consider the symmetric inverse monoid ℐn of an n-element chain and its inverse submonoids ℐn, ℐn, ℐn and ℘ℐn of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations, respectively, and give descriptions of their Abelian kernels relative to decidable pseudovarieties of Abelian groups.

A presentation for the monoid of uniform block permutations

The monoid n of uniform block permutations is the factorisable inverse monoid which arises from the natural action of the symmetric group on the join semilattice of equivalences on an n-set; it has been described in the literature as the factorisable part of the dual symmetric inverse monoid. The present paper gives and proves correct a monoid presentation forn. The methods involved make use of a general criterion for a monoid generated by a group and an idempotent to be inverse, the structure of factorisable inverse monoids, and presentations of the symmetric group and the join semilattice of equivalences on an n-set.

Dual symmetric inverse monoids and representation theory

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.