%0 Journal Article
%T On the Monoid of All Order Preserving Full Contractions with a Fixed Set
%A Muhammad Mansur Zubairu
%A Zulyadaini D. Lawan
%J Open Journal of Discrete Mathematics
%P 55-71
%@ 2161-7643
%D 2025
%I Scientific Research Publishing
%R 10.4236/ojdm.2025.153004
%X In this paper, we consider the monoid <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>Fix</mtext><mi mathvariant='script'>O</mi><mi mathvariant='script'>C</mi><mi mathvariant='script'>T</mi><mrow><mo>(</mo>
   <mrow>
    <mrow><mo>[</mo> <mi>n</mi> <mo>]</mo></mrow><mo>,</mo><mi>D</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

 of all order-preserving full contraction mappings that fix a subset say <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>D</mi>
</math>

 of a finite <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>n</mi>
</math>

-element chain <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>{</mo> <mrow>
   <mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>&#x22EF;</mo><mo>,</mo><mi>n</mi></mrow> <mo>}</mo></mrow></mrow>
</math>

. We characterize regularity, Green&#8217;s relations, and starred Green&#8217;s relations, and show that this monoid is left adequate. Furthermore, we determine the cardinality of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>Fix</mtext><mi mathvariant='script'>O</mi><mi mathvariant='script'>C</mi><mi mathvariant='script'>T</mi><mrow><mo>(</mo>
   <mrow>
    <mrow><mo>[</mo> <mi>n</mi> <mo>]</mo></mrow><mo>,</mo><mi>D</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

, the ranks of its two-sided ideals, and demonstrate that the ranks of the two-sided ideals and their corresponding Rees quotients are equal. Moreover, we deduce the rank of the monoid <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>Fix</mtext><mi mathvariant='script'>O</mi><mi mathvariant='script'>C</mi><mi mathvariant='script'>T</mi><mrow><mo>(</mo>
   <mrow>
    <mrow><mo>[</mo> <mi>n</mi> <mo>]</mo></mrow><mo>,</mo><mi>D</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

.
%K Contraction Map
%K Order Preserving
%K Fixed Set
%K Rank
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=143844