Generating the full transformation semigroup using order preserving mappings

by James Mitchell

For a semigroup $S$, the `classical' idea of rank is concerned with finding minimum size generating sets for $S$. When working with a finitely generated semigroup $S$ determining the rank of $S$ is a natural consideration. However, for an uncountable semigroup $S$ the rank of $S$ is $\vert S\vert$, and so the classical notion of rank provides us with no information. We introduce a different rank property which allows us to `measure', from a certain perspective, a given semigroup with respect to some distinguished subsemigroups. For a semigroup $S$, if $A\subseteq S$ then we call the minimum cardinality of a set $B$ such that

$$\langle\:A\cup B\;\rangle=S,$$

the relative rank of $S$ modulo $A$. In this talk we shall consider the relative rank of the full transformation semigroup $\mathcal{T}_{X}$, over an infinite set $X$, modulo various standard subsemigroups. In particular, we give the relative rank of $\mathcal{T}_{X}$ modulo the semigroup of all order preserving maps, for some infinite ordered sets $X$.