We study the algebraic properties of the semigroup $\mathscr{O\!\!I\!}_n(L)$ of finite partial order isomorphisms of rank $\leq n$ of an infinite linearly ordered set $(L,\leqslant)$. In particular, its idempotents, natural partial order, and Green's relation on $\mathscr{O\!\!I\!}_n(L)$ are described. It is proved that the semigroup $\mathscr{O\!\!I\!}_n(L)$ is stable and contains a dense series of ideals, and that all congruences on the semigroup $\mathscr{O\!\!I\!}_n(L)$ are Riesz congruences.

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