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