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Partial transformation monoids preserving a uniform partition

Abstract

The objective of this paper is to study the monoid of all partial transformations of a finite set that preserve a uniform partition. In addition to proving that this monoid is a quotient of a wreath product with respect to a congruence relation, we show that it is generated by 5 generators, we compute its order and determine a presentation on a minimal generating set.

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Acknowledgments

Cicalò gratefully acknowledges support of FCT and PIDDAC, within the Project PTDC/MAT/69514/2006 of CAUL, and support of the Regione Sardegna, within the Project Master & Back, PR-MAB-A2009-837. She also thanks the Department of Mathematics of the University of Trento for its hospitality from 2011. Fernandes gratefully acknowledges support of FCT and PIDDAC, within the Projects ISFL-1-143 and PTDC/MAT/69514/2006 of CAUL. Schneider was supported by the research funds PTDC/MAT/101993/2008 (FCT), 302660/2013-5 (CNPq, Produtividade em Pesquisa), 475399/2013-7 (CNPq, Universal), and by APQ-00452-13 (Fapemig, Universal).

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Correspondence to Csaba Schneider.

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Communicated by Mikhail Volkov.

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Cicalò, S., Fernandes, V.H. & Schneider, C. Partial transformation monoids preserving a uniform partition. Semigroup Forum 90, 532–544 (2015). https://doi.org/10.1007/s00233-014-9629-5

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Keywords

  • Transformation monoids
  • Uniform partitions
  • Ranks
  • Presentations
  • Partial transformations
  • Wreath products