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Infinite partition monoids

James East

Centre for Research in Mathematics, School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith NSW 2751, Australia

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https://doi.org/10.1142/S0218196714500209Cited by:13 (Source: Crossref)

Abstract

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

Keywords:

AMSC: 20M20, 20M17

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