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

Title & Authors
A NOTE ON BILATERAL SEMIDIRECT PRODUCT DECOMPOSITIONS OF SOME MONOIDS OF ORDER-PRESERVING PARTIAL PERMUTATIONS
Fernandes, Vitor H.; Quinteiro, Teresa M.;

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