# On the monoid of cofinite partial isometries of $\mathbb{N}^n$ with the
usual metric

Research paper by **Oleg Gutik, Anatolii Savchuk**

Indexed on: **21 Sep '19**Published on: **19 Sep '19**Published in: **arXiv - Mathematics - Group Theory**

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#### Abstract

In this paper we study the structure of the monoid
$\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th
power of the set of positive integers $\mathbb{N}$ with the usual metric for a
positive integer $n\geqslant 2$. We describe the elements of the monoid
$\mathbf{I}\mathbb{N}_{\infty}^n$ as partial transformation of $\mathbb{N}^n$,
the group of units and the subset of idempotents of the semigroup
$\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's
relations on $\mathbf{I}\mathbb{N}_{\infty}^n$. In particular we show that the
quotient semigroup
$\mathbf{I}\mathbb{N}_{\infty}^n/\mathfrak{C}_{\textsf{mg}}$, where
$\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence on
$\mathbf{I}\mathbb{N}_{\infty}^n$, is isomorphic to the symmetric group
$\mathscr{S}_n$ and $\mathscr{D}=\mathscr{J}$ in
$\mathbf{I}\mathbb{N}_{\infty}^n$. Also, we prove that for any integer
$n\geqslant 2$ the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$ is isomorphic to
the semidirect product
${\mathscr{S}_n\ltimes_\mathfrak{h}(\mathscr{P}_{\infty}(\mathbb{N}^n),\cup)}$
of the free semilattice with the unit
$(\mathscr{P}_{\infty}(\mathbb{N}^n),\cup)$ by the symmetric group
$\mathscr{S}_n$.