The Rank of the Endomorphism Monoid of a Partition
Group Theory
2008-07-09 v1 Rings and Algebras
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.
Keywords
Cite
@article{arxiv.0807.1214,
title = {The Rank of the Endomorphism Monoid of a Partition},
author = {Joao Araujo and Csaba Schneider},
journal= {arXiv preprint arXiv:0807.1214},
year = {2008}
}
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11 pages