On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set - Gutik, O.; Shchypel, M. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 512.536We study the topologization of the semigroup $\mathscr{O\!\!I}\!_n(L)$ of finite partial order isomorphisms of bounded rank of an infinite linear ordered set $(L,\leqslant).$ In particular, we show that every $T_1$ left-topological (right-topological) semigroup $\mathscr{O\!\!I}\!_n(L)$ is an Urysohn, functionally  Hausdorff, totally disconnected, and scattered space. It  is proved that, on the semigroup $\mathscr{O\!\!I}\!_n(L),$ there exists a unique Hausdorff countably compact (pseudocompact) shift-continuous topology,  which is compact, and that the Bohr compactification of the Hausdorff topological semigroup $\mathscr{O\!\!I}\!_n(L)$ is the trivial semigroup.