title: PartialOrdering, StrictWeakOrdering, TotalOrdering, what's the main difference in application
body: <p><a href="http://www.sgi.com/tech/stl/LessThanComparable.html" rel="noreferrer"> [SGI official document]</a></p>
  
  <blockquote>
    <p>Because of irreflexivity and transitivity, operator&lt; always satisfies
    the definition of a partial ordering. The definition of a strict weak
    ordering is stricter, and the definition of a total ordering is
    stricter still.</p>
  </blockquote>
  
  <p>And I also read the definition of strict weak ordering in the document: <a href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html" rel="noreferrer">StrictWeakOrdering</a></p>
  
  <blockquote>
    <p>The first three axioms, irreflexivity, antisymmetry, and transitivity,
    are the definition of a partial ordering; transitivity of equivalence
    is required by the definition of a strict weak ordering. A total
    ordering is one that satisfies an even stronger condition: equivalence
    must be the same as equality.</p>
  </blockquote>
  
  <p>I am not quite sure about these definition. Some main questions:</p>
  
  <p><strong>1.Is <em>partial ordering</em> implicitly define a equivalence?</strong></p>
  
  <p><strong>2.What about <em>strict weak ordering</em> and <em>total ordering</em>?</strong></p>
  
  <p><strong>3.STL require a strict weak ordering in sort algorithms, why isn't partial ordering or total ordering?</strong>
  For this question, I've read some textbooks that prove a valid comparing rules by proving the rule satisfy three axioms: irreflexivity, antisymmetry, transitivity which is the definition for partial ordering, and the document refer that operator&lt; always satisfies this definition, So why can't we just compare objects using partial ordering, or equivalently, using operator