Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on bilateral semidirect product decompositions of some monoids of
  order-preserving partial permutations   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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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga