%0 Journal Article
%T Research on Nilpotent Groups Based on Fuzzy Hypergroups
%A Yi Yan
%A Yan Yan
%A Xinru Shang
%J Journal of Applied Mathematics and Physics
%P 148-156
%@ 2327-4379
%D 2025
%I Scientific Research Publishing
%R 10.4236/jamp.2025.131006
%X In this paper, by constructing the smallest equivalence relation <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>&#x03B8;</mi>
   <mo>&#x2217;</mo>
  </msup>
  </mrow>
</math>

 on a finite fuzzy hypergroup <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>H</mi>
</math>

, the quotient group (the set of equivalence classes) <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mi>H</mi><mo>/</mo><mrow>
   <msup>
    <mi>&#x03B8;</mi>
    <mo>&#x2217;</mo>
   </msup>
   </mrow></mrow>
  </mrow>
</math>

 is a nilpotent group, and the nilpotent group is characterized by the strong fuzzy regularity of the equivalence relation. Finally, the concept of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>&#x03B8;</mi>
</math>

-part of fuzzy hypergroup is introduced to determine the necessary and sufficient condition for the equivalence relation <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>&#x03B8;</mi>
</math>

 to be transitive.
%K Fuzzy Hypergroup
%K Equivalence Relation
%K Strong Fuzzy Regular Relation
%K Nilpotent Group
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=139930