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<Head>
<Title>Strict Weak Ordering</Title>
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<H1>Strict Weak Ordering</H1>
<Table CellPadding=0 CellSpacing=0 width=100%>
<TR>
<TD Align=left><Img src = "functors.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD>
<TD Align=right><Img src = "concept.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD>
</TR>
<TR>
<TD Align=left VAlign=top><b>Category</b>: functors</TD>
<TD Align=right VAlign=top><b>Component type</b>: concept</TD>
</TR>
</Table>
<h3>Description</h3>
A Strict Weak Ordering is a <A href="BinaryPredicate.html">Binary Predicate</A> that compares two
objects, returning <tt>true</tt> if the first precedes the second. This
predicate must satisfy the standard mathematical definition of
a <i>strict weak ordering</i>. The precise requirements are stated
below, but what they roughly mean is that a Strict Weak Ordering has
to behave the way that "less than" behaves: if <tt>a</tt> is less than <tt>b</tt>
then <tt>b</tt> is not less than <tt>a</tt>, if <tt>a</tt> is less than <tt>b</tt> and <tt>b</tt> is less
than <tt>c</tt> then <tt>a</tt> is less than <tt>c</tt>, and so on.
<h3>Refinement of</h3>
<A href="BinaryPredicate.html">Binary Predicate</A>
<h3>Associated types</h3>
<Table border>
<TR>
<TD VAlign=top>
First argument type
</TD>
<TD VAlign=top>
The type of the Strict Weak Ordering's first argument.
</TD>
</TR>
<TR>
<TD VAlign=top>
Second argument type
</TD>
<TD VAlign=top>
The type of the Strict Weak Ordering's second argument. The first
argument type and second argument type must be the same.
</TD>
</TR>
<TR>
<TD VAlign=top>
Result type
</TD>
<TD VAlign=top>
The type returned when the Strict Weak Ordering is called. The result type
must be convertible to <tt>bool</tt>.
</TD>
</tr>
</table>
<h3>Notation</h3>
<Table>
<TR>
<TD VAlign=top>
<tt>F</tt>
</TD>
<TD VAlign=top>
A type that is a model of Strict Weak Ordering
</TD>
</TR>
<TR>
<TD VAlign=top>
<tt>X</tt>
</TD>
<TD VAlign=top>
The type of Strict Weak Ordering's arguments.
</TD>
</TR>
<TR>
<TD VAlign=top>
<tt>f</tt>
</TD>
<TD VAlign=top>
Object of type <tt>F</tt>
</TD>
</TR>
<TR>
<TD VAlign=top>
<tt>x</tt>, <tt>y</tt>, <tt>z</tt>
</TD>
<TD VAlign=top>
Object of type <tt>X</tt>
</TD>
</tr>
</table>
<h3>Definitions</h3>
<UL>
<LI>
Two objects <tt>x</tt> and <tt>y</tt> are <i>equivalent</i> if both <tt>f(x, y)</tt> and
<tt>f(y, x)</tt> are false. Note that an object is always (by the
irreflexivity invariant) equivalent to itself.
</UL>
<h3>Valid expressions</h3>
None, except for those defined in the <A href="BinaryPredicate.html">Binary Predicate</A> requirements.
<h3>Expression semantics</h3>
<Table border>
<TR>
<TH>
Name
</TH>
<TH>
Expression
</TH>
<TH>
Precondition
</TH>
<TH>
Semantics
</TH>
<TH>
Postcondition
</TH>
</TR>
<TR>
<TD VAlign=top>
Function call
</TD>
<TD VAlign=top>
<tt>f(x, y)</tt>
</TD>
<TD VAlign=top>
The ordered pair <tt>(x,y)</tt> is in the domain of <tt>f</tt>
</TD>
<TD VAlign=top>
Returns <tt>true</tt> if <tt>x</tt> precedes <tt>y</tt>, and <tt>false</tt> otherwise
</TD>
<TD VAlign=top>
The result is either <tt>true</tt> or <tt>false</tt>
</TD>
</tr>
</table>
<h3>Complexity guarantees</h3>
<h3>Invariants</h3>
<Table border>
<TR>
<TD VAlign=top>
Irreflexivity
</TD>
<TD VAlign=top>
<tt>f(x, x)</tt> must be <tt>false</tt>.
</TD>
</TR>
<TR>
<TD VAlign=top>
Antisymmetry
</TD>
<TD VAlign=top>
<tt>f(x, y)</tt> implies <tt>!f(y, x)</tt>
</TD>
</TR>
<TR>
<TD VAlign=top>
Transitivity
</TD>
<TD VAlign=top>
<tt>f(x, y)</tt> and <tt>f(y, z)</tt> imply <tt>f(x, z)</tt>.
</TD>
</TR>
<TR>
<TD VAlign=top>
Transitivity of equivalence
</TD>
<TD VAlign=top>
Equivalence (as defined above) is transitive: if <tt>x</tt> is equivalent
to <tt>y</tt> and <tt>y</tt> is equivalent to <tt>z</tt>, then <tt>x</tt> is equivalent to <tt>z</tt>.
(This implies that equivalence does in fact satisfy the mathematical
definition of an equivalence relation.) <A href="#1">[1]</A>
</TD>
</tr>
</table>
<h3>Models</h3>
<UL>
<LI>
<tt><A href="less.html">less</A><int></tt>
<LI>
<tt><A href="less.html">less</A><double></tt>
<LI>
<tt><A href="greater.html">greater</A><int></tt>
<LI>
<tt><A href="greater.html">greater</A><double></tt>
</UL>
<h3>Notes</h3>
<P><A name="1">[1]</A>
The first three axioms, irreflexivity, antisymmetry, and
transitivity, are the definition of a <i>partial ordering</i>;
transitivity of equivalence is required by the definition of a
<i>strict weak ordering</i>. A <i>total ordering</i> is one that satisfies
an even stronger condition: equivalence must be the same as equality.
<h3>See also</h3>
<A href="LessThanComparable.html">LessThan Comparable</A>, <tt><A href="less.html">less</A></tt>, <A href="BinaryPredicate.html">Binary Predicate</A>,
<A href="functors.html">function objects</A>
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