The semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

We study algebraic properties of the semigroup ${\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}}_n(L)$ of finite partial order isomorphisms of the rank $\le n$ of an infinite linearly ordered set $(L,⩽)$. In particular we describe its idempotents, the natural partial order and Green’s relations on ${\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}}_n(L)$. It is proved that the semigroup ${\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}}_n(L)$ is stable and it contains tight ideal series. Moreover, we show that the semigroup ${\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}}_n(L)$ admits only Rees’ congruences and every its homomorphic image is a semigroup with tight ideal series.