The rank of the endomorphism monoid of a uniform partition |
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| Authors: | João Araújo Csaba Schneider |
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| Institution: | (1) Universidade Aberta, R. Escola Politécnica 147, 1269-001 Lisbon, Portugal;(2) Informatics Research Laboratory, Computer and Automation Research Institute, P.O. Box 63, 1518 Budapest, Hungary;(3) Centro de álgebra, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisbon, Portugal |
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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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| Keywords: | Transformation semigroups Rank Relative rank Wreath product Symmetric groups |
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