Abstract
The partition monoid \(\mathcal P_n\) is known to be minimally \(4\)-generated (for \(n\ge 3\)). Modulo some small values of \(n\), we show that: (1) \(\mathcal P_n\) embeds in a \(3\)-generator subsemigroup of \(\mathcal P_{n+1}\); (2) \(\mathcal P_n\) does not embed in a \(2\)-generator subsemigroup of \(\mathcal P_{n+1}\); and (3) \(\mathcal P_n\) embeds in a \(2\)-generator subsemigroup of \(\mathcal P_{n+3}\). A consequence of (3) is that every finite semigroup embeds in a finite \(2\)-generator regular \(*\)-semigroup.
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Notes
Unless otherwise stated, generation will always be within the variety of semigroups.
The monoid \(\mathcal P_1\) is already \(2\)-generated. The author does not currently know whether \(\mathcal P_2\) (which is \(3\)-generated) embeds in a \(2\)-generator subsemigroup of \(\mathcal P_3\), but suspects it does not.
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East, J. Partition monoids and embeddings in 2-generator regular \(*\)-semigroups. Period Math Hung 69, 211–221 (2014). https://doi.org/10.1007/s10998-014-0055-y
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DOI: https://doi.org/10.1007/s10998-014-0055-y