Infinite partition monoids

Abstract

Let 𝒫X and 𝒮X be the partition monoid and symmetric group on an infinite set X. We show that 𝒫X may be generated by 𝒮X together with two (but no fewer) additional partitions, and we classify the pairs α, β ∈ 𝒫X for which 𝒫X is generated by 𝒮X ∪ {α, β}. We also show that 𝒫X may be generated by the set ℰX of all idempotent partitions together with two (but no fewer) additional partitions. In fact, 𝒫X is generated by ℰX ∪ {α, β} if and only if it is generated by ℰX ∪ 𝒮X ∪ {α, β}. We also classify the pairs α, β ∈ 𝒫X for which 𝒫X is generated by ℰX ∪ {α, β}. Among other results, we show that any countable subset of 𝒫X is contained in a 4-generated subsemigroup of 𝒫X, and that the length function on 𝒫X is bounded with respect to any generating set.