The Rank of the Endomorphism Monoid of a Partition

Details The Rank of the Endomorphism Monoid of a Partition The Rank of the Endomorphism Monoid of a Partition

2008-07-08

arxiv.org/abs/0807.1214v1

Mathematics

Group Theory

11 pages

Scientific paper

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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