Tue, 9 September 2025, 1:25 pm MT
Let S be a semigroup, and suppose that the full relation SxS is finitely generated as a right congruence. When S is a group this happens if and only if S is finitely generated, but in general there are many more examples, of arbitrarily large cardinalities. Let X be a finite generating set for SxS. 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. $\leq 4$. 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.