Mathematics – Rings and Algebras
Scientific paper
2011-04-11
Mathematics
Rings and Algebras
18 pages
Scientific paper
We classify the maximal subsemigroups of the semigroup $\Omega^\Omega$ of all mappings on an infinite set $\Omega$ that contain one of the following groups: the symmetric group on $\Omega$, the setwise stabilizer of a non-empty finite subset of $\Omega$, the stabilizer of a finite partition of $\Omega$, or the stabilizer of an ultrafilter on $\Omega$. If $G$ is any of these groups, then we also characterise the mappings $f,g\in \Omega^\Omega$ such that the semigroup $\genset{G, f, g}$ generated by $G\cup \{f,g\}$ equals $\Omega^\Omega$. We also show that the setwise stabiliser of a non-empty finite set, the almost stabiliser of a finite partition, and the stabiliser of an ultrafilter are maximal subsemigroups of the symmetric group.
East Jack
Mitchell James D.
Péresse Y.
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