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We study algebraic properties of the semigroup $\\mathscr{I\\!\\!P\\!F}(\\mathbb{N}^n)$. In particular, we show that $\\mathscr{I\\!\\!P\\!F}(\\mathbb{N}^n)$ is a bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\\mathscr{I\\!\\!P\\!F}(\\mathbb{N}^n)$ and its maximal subgroups. 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