On the diameter of semigroups of transformations and partitions

Abstract

For a semigroup $S$ whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right-$F{P}_1$), the right diameter of $S$ is a parameter that expresses how ‘far apart’ elements of $S$ can be from each other, in a certain sense. To be more precise, for each finite generating set $U$ for the universal right congruence on $S$, we have a metric space $(S,d_U)$ where $d_U(a,b)$ is the minimum length of derivations for $(a,b)$ as a consequence of pairs in $U$; the right diameter of $S$ with respect to $U$ is the diameter of this metric space. The right diameter of $S$ is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set $X$ has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on $X$, of all partial transformations on $X$, and of all full transformations on $X$, as well as the partition and partial Brauer monoids on $X$, have right diameter 1 and left diameter 1. The symmetric inverse monoid on $X$ has right diameter 2 and left diameter 2. The monoid of all injective mappings on $X$ has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on $X$) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on $X$ has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.