The rank of the endomorphism monoid of a uniform partition
The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invaria...
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| Formato: | article |
| Idioma: | eng |
| Publicado em: |
2011
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| Texto completo: | http://hdl.handle.net/10400.2/1999 |
| País: | Portugal |
| Oai: | oai:repositorioaberto.uab.pt:10400.2/1999 |
| Resumo: | The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question. |
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