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<p>We study ideals and filters of posets and of pseudocomplemented posets and show a version of the Separation Theorem, known for ideals and filters in lattices and semilattices, within this general setting. We extend the concept of a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(*\)</EquationSource> </InlineEquation>-ideal already introduced by Rao for pseudocomplemented distributive lattices and by Talukder, Chakraborty and Begum for pseudocomplemented semilattices to pseudocomplemented posets. We derive several important properties of such ideals. Especially, we explain connections between prime filters, ultrafilters, filters satisfying the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(*\)</EquationSource> </InlineEquation>-condition and dense elements. Finally, we prove a Separation Theorem for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(*\)</EquationSource> </InlineEquation>-ideals.</p>

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Filters and ideals in pseudocomplemented posets

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月31日

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  • Ivan Chajda,
  • Helmut Länger

摘要

We study ideals and filters of posets and of pseudocomplemented posets and show a version of the Separation Theorem, known for ideals and filters in lattices and semilattices, within this general setting. We extend the concept of a \(*\) -ideal already introduced by Rao for pseudocomplemented distributive lattices and by Talukder, Chakraborty and Begum for pseudocomplemented semilattices to pseudocomplemented posets. We derive several important properties of such ideals. Especially, we explain connections between prime filters, ultrafilters, filters satisfying the \(*\) -condition and dense elements. Finally, we prove a Separation Theorem for \(*\) -ideals.