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<p>The main aim of this paper is to present a Stone-type topological duality for posets, that is, we develop a categorical duality between the category of posets and a category of certain topological spaces. The principal tool to achieve this goal is the notion of <i>ud</i>-sets. More precisely, we introduce the notion of <i>ud</i>-sets, and then we use this notion to build a duality between posets without greatest elements and <i>PSK</i>-spaces, and a duality between posets with greatest elements and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(PSK^{\top }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>S</mi> <msup> <mi>K</mi> <mi>⊤</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>-spaces. Furthermore, we apply this dual equivalence to obtain Stone-type topological dualities for dcpos and complete lattices, respectively.</p>

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A Stone-type Topological Duality for Posets

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Jing Lu,
  • Bin Zhao

摘要

The main aim of this paper is to present a Stone-type topological duality for posets, that is, we develop a categorical duality between the category of posets and a category of certain topological spaces. The principal tool to achieve this goal is the notion of ud-sets. More precisely, we introduce the notion of ud-sets, and then we use this notion to build a duality between posets without greatest elements and PSK-spaces, and a duality between posets with greatest elements and \(PSK^{\top }\) P S K ⊤ -spaces. Furthermore, we apply this dual equivalence to obtain Stone-type topological dualities for dcpos and complete lattices, respectively.