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<p>In this article, we give algebraic characterizations of <i>U</i>-frames in terms of ring-theoretic properties of the ring <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_888_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {R}L\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> of real-valued continuous functions on a completely regular frame <i>L</i>. We show that a frame is a <i>U</i>-frame if and only if it is an <i>F</i>-frame and its Čech–Stone compactification is zero-dimensional. We will also introduce frames that are finitely a <i>U</i>-frame and we will characterize them in terms of ring-theoretic properties in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_888_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {R}L\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Characterizations of U-frames and frames that are finitely a U-frame

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月25日

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  • Batsile Tlharesakgosi

摘要

In this article, we give algebraic characterizations of U-frames in terms of ring-theoretic properties of the ring \(\mathcal {R}L\) R L of real-valued continuous functions on a completely regular frame L. We show that a frame is a U-frame if and only if it is an F-frame and its Čech–Stone compactification is zero-dimensional. We will also introduce frames that are finitely a U-frame and we will characterize them in terms of ring-theoretic properties in \(\mathcal {R}L\) R L .