Rank and Status in Semigroup Theory

John M. Howie
Mathematical Institute
University of St. Andrews
St. Andrews
KY16 9SS, Scotland, U.K.
e-mail: jmh@st-andrews.ac.uk

Abstract:

Associated with every generating set $A$ for a finite semigroup $S$ is a parameter $\Delta_{A}$, defined as the least $n$ for which

$$A \cup A^{2} \cup \cdots \cup A^{n} = S .$$

Generally speaking, a larger $A$ implies a smaller $\Delta_{A}$, and the product $\vert A\vert\Delta_{A}$ is a crude measure of the ` efficiency' of the generating set. The status $\mathrm{Stat}(S)$ of $S$ is defined by

$$\mathrm{Stat}(S) = \min\{ \vert A\vert\Delta_{A} : A \subseteq S , \; \langle A\rangle = S\} .$$

The lecture, whgich reports on work done jointly with Alessandra Cherubini and Brunetto Piochi, explores this idea.