Infinite partition monoids

Abstract

Let ${\mathcal{P}}_X$ and ${\mathcal{S}}_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that ${\mathcal{P}}_X$ may be generated by ${\mathcal{S}}_X$ together with two (but no fewer) additional partitions, and we classify the pairs $\alpha ,\beta \in {\mathcal{P}}_X$ for which ${\mathcal{P}}_X$ is generated by ${\mathcal{S}}_X\cup \{\alpha ,\beta \}$. We also show that ${\mathcal{P}}_X$ may be generated by the set ${\mathcal{E}}_X$ of all idempotent partitions together with two (but no fewer) additional partitions. In fact, ${\mathcal{P}}_X$ is generated by ${\mathcal{E}}_X\cup \{\alpha ,\beta \}$ if and only if it is generated by ${\mathcal{E}}_X\cup {\mathcal{S}}_X\cup \{\alpha ,\beta \}$. We also classify the pairs $\alpha ,\beta \in {\mathcal{P}}_X$ for which ${\mathcal{P}}_X$ is generated by ${\mathcal{E}}_X\cup \{\alpha ,\beta \}$. Among other results, we show that any countable subset of ${\mathcal{P}}_X$ is contained in a $4$-generated subsemigroup of ${\mathcal{P}}_X$, and that the length function on ${\mathcal{P}}_X$ is bounded with respect to any generating set.

Cited by in corpus (2)

• Enumeration of idempotents in diagram semigroups and algebras
• Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones