An endomorphism of the free monoid A∗ is invertible if it is injective and extends to an automorphism of the free group generated by A. A simple example: the endomorphism that leaves all generators A invariant except one, say a, which is mapped to ba for some other generator b. We give a monoid presentation for the submonoid generated by all such endomorphisms when a and b are taken arbitrarily...

We described in [7] a monoid b G acting on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid with unit group a Kac-Moody group G. Now we find natural extensions of the action of the Kac-Moody group G on its building Ω to actions of the monoid b G on Ω. These extensions are partly motivated by represen...

We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition order. We apply our result to establish that the commutative monoid of weakly normed processes modulo...

The aim of this paper is sketch a theory of divisibility and factorisation in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely topologised topological monoids. In particular, we define the topological factorisation monoid, a generalisation of the factorisation monoid for algebraic monoids, and...

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E(−). Our description is as a graded and completed version of a Tall-Wraith monoid. The E-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the...

We show how the categorial approach to inverse monoids can be described as a certain endofunctor (which we call the partialization functor) of some category. In this paper we show that this functor can be used to obtain several recently defined inverse monoids, and use it to define a new object, which we call the inverse braid-permutation monoid. A presentation for this monoid is obtained. Fina...

This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M . In fact, our notion is that of T -proper, where T is a submonoid of M . We show that any left adequate monoid M has an X∗proper cover for some set X , that i...

Ehresmann monoids form a variety of biunary monoids, that is, monoids equipped with two basic unary operations, the images of which coincide and form a semilattice of projections. The monoid of binary relations BX on any setX with unary operations of domain and range is Ehresmann. Inverse monoids, regarded as biunary submonoids of BX via the Wagner-Preston representation theorem, are therefore ...

In this paper, we first consider n × n upper-triangular matrices with entries in a given semiring k. Matrices of this form with invertible diagonal entries form a monoid Bn(k). We show that Bn(k) splits as a semidirect product of the monoid of unitriangular matrices Un(k) by the group of diagonal matrices. When the semiring is a field, Bn(k) is actually a group and we recover a well-known resul...

In this paper we will be on erned with the possibility of embedding a nite semigroup S into a nite 2-generated semigroup T that shares properties with S. In parti ular we show that any nite orthodox semigroup S may be embedded in a nite orthodox semigroup T generated by two group elements and that any nite orthodox monoid S1 may be embedded as a semigroup into a nite 2-generated orthodox monoid...