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<p>We study the problem of topologization of a semigroup <i>𝒪ℐ</i><sub><i>n</i></sub>(<i>L</i>) of finite partial-order isomorphisms of bounded rank of an infinite linearly ordered set (<i>L</i>,<i>≤</i>)<i>.</i> In particular, it is shown that every <i>T</i><sub>1</sub> lefttopological (right-topological) semigroup <i>𝒪ℐ</i><sub><i>n</i></sub>(<i>L</i>) is an Urysohn, functionally Hausdorff, totally disconnected, and scattered space. It is also proved that, on the semigroup <i>𝒪ℐ</i><sub><i>n</i></sub>(<i>L</i>)<i>,</i> there exists a unique Hausdorff countably compact (pseudocompact) shift-continuous topology, which is compact, and that the Bohr compactification of the Hausdorff topological semigroup <i>𝒪ℐ</i><sub><i>n</i></sub>(<i>L</i>) is the trivial semigroup.</p>

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On Compact Topologies on the Semigroup of Finite Partial-Order Isomorphisms of Bounded Rank of an Infinite Linearly Ordered Set

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • Oleg Gutik,
  • Maksym Shchypel

摘要

We study the problem of topologization of a semigroup 𝒪ℐn(L) of finite partial-order isomorphisms of bounded rank of an infinite linearly ordered set (L,≤). In particular, it is shown that every T1 lefttopological (right-topological) semigroup 𝒪ℐn(L) is an Urysohn, functionally Hausdorff, totally disconnected, and scattered space. It is also proved that, on the semigroup 𝒪ℐn(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 𝒪ℐn(L) is the trivial semigroup.