University of Sydney Algebra Seminar

Nik Ruškuc

Friday 13 November, 12-1pm, in SMRI Seminar Room (Macleay Building A12 Room 301)

Congruence diameter of semigroups

Actions are a fundamental concept in semigroup theory. They play the role analogous to modules in rings and permutation representations for groups. A closely related concept is that of a right (or left) congruence - mirroring the role ideals and subgroups. In this talk I will discuss an intriguing finiteness condition - that the full relation \(SxS\) on a semigroup \(S\) is finitely generated as a right congruence - and a natural notion of diameter accompanying it. When \(S\) is a group the condition is equivalent to finite generation of \(S\) itself, and the diameter being finite is equivalent to \(S\) itself being finite. However, in general, there are many 'weird' interesting examples, many of them 'very infinite'. So, let \(S\) be a semigroup, and suppose that \(X\) is a finite generating set for \(S \times S\). Then, for any two elements \(s,t\) in \(S\), there is a finite chain of \(X\)-transformations connecting \(s\) and \(t\). The longest such chain is the \(X\)-diameter of \(S\), and the smallest \(X\)-diameter when \(X\) ranges over all finite generating sets is the (right congruence) diameter of \(S\), denoted \(D(S)\). This parameter can be finite or infinite. It turns out that it is finite for many classical semigroups of transformations, linear transformations and partitions, and is then very small, i.e. at most \(4\). Usually some intriguing combinatorics is involved in these results, but the underlying reasons for this phenomenon remain mysterious. The new results presented in this talk are due to various subsets of J. East, V. Gould, C. Miller, T. Quinn-Gregson and myself.