Bulletin of the Korean Mathematical Society (대한수학회보)

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A NOTE ON BILATERAL SEMIDIRECT PRODUCT DECOMPOSITIONS OF SOME MONOIDS OF ORDER-PRESERVING PARTIAL PERMUTATIONS

  • Fernandes, Vitor H. (Departamento de Matematica, Faculdade de Ciencias e Tecnologia, Universidade NOVA de Lisboa, Centro de Algebra da Universidade de Lisboa) ;
  • Quinteiro, Teresa M. (Instituto Superior de Engenharia de Lisboa, Centro de Algebra da Universidade de Lisboa)
  • Received : 2015.02.26
  • Published : 2016.03.31
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Abstract

In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,{\ldots},n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of $\mathcal{ODP}_n$ and $\mathcal{C}_2$.

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References

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