The Rank of the Endomorphism Monoid of a Partition

Joao Araujo; Csaba Schneider

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arXiv · 0807.1214

The Rank of the Endomorphism Monoid of a Partition

Joao Araujo·Csaba Schneider

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Abstract

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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BibTeX RIS

Joao Araujo, Csaba Schneider. 2008-07-08. The Rank of the Endomorphism Monoid of a Partition. https://arxiv.org/abs/0807.1214

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