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<p>We initiate the algebraic study of the semigroup of one-to-one order-preserving partial contraction mappings of a totally ordered set <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1259_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{1,2,\ldots ,n\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, which we denote by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1259_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {OCI}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">OCI</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. In particular, we characterise the Green’s relations and their starred analogues in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1259_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {OCI}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">OCI</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. We also compute the rank of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1259_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {OCI}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">OCI</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1259_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(2n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the semigroup of finite order-preserving partial injective contraction mappings

  • 期刊论文

  • 发表日期：

    2025年2月4日

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  • F. Al-Kharousi,
  • G. U. Garba,
  • M. J. Ibrahim,
  • A. T. Imam,
  • A. Umar

摘要

We initiate the algebraic study of the semigroup of one-to-one order-preserving partial contraction mappings of a totally ordered set \(\{1,2,\ldots ,n\}\) { 1 , 2 , … , n } , which we denote by \(\mathcal {OCI}_{n}\) OCI n . In particular, we characterise the Green’s relations and their starred analogues in \(\mathcal {OCI}_{n}\) OCI n . We also compute the rank of \(\mathcal {OCI}_{n}\) OCI n as \(2n-1\) 2 n - 1 .