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| <Head> |
| <Title>Monoid</Title> |
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| <BR Clear> |
| |
| <H1><A NAME="concept:Monoid"></A> |
| Monoid |
| </H1> |
| |
| A <i>Monoid</i> is a concept that describes a simple kind of algebraic |
| system. A <b>monoid</b> consists of a set of elements <i>S</i>, a |
| binary operation, and an identity element. The C++ representation of a |
| monoid consists of a function object that implements the binary |
| operation, a set of objects that represent the elements of <i>S</i>, |
| and an object that represents the identity element. |
| |
| |
| <H3>Refinement of</H3> |
| |
| The element type must be a model of <a |
| href="../../utility/Assignable.html">Assignable</a> and <a |
| href="../../utility/CopyConstructible.html">CopyConstructible</a>. |
| The function object type must be a model of <a |
| href="http://www.sgi.com/tech/stl/BinaryFunction.html">BinaryFunction</a>. |
| |
| <h3>Notation</h3> |
| |
| <Table> |
| <TR> |
| <TD><tt>X</tt></TD> |
| <TD>Is the element type of the Monoid.</TD> |
| </TR> |
| |
| <TR> |
| <TD><tt>a, b</tt></TD> |
| <TD>are objects of type <tt>X</tt>.</TD> |
| </TR> |
| |
| <TR> |
| <TD><tt>op</tt></TD> |
| <TD>Is the function object implementing the Monoid operation.</TD> |
| </TR> |
| |
| <TR> |
| <TD><tt>i</tt></TD> |
| <TD>is an object of type <tt>X</tt> and is the identity element |
| for the Monoid.</TD> |
| </TR> |
| |
| |
| </table> |
| |
| <h3>Valid Expressions</h3> |
| |
| <Table border> |
| |
| <tr> |
| <td><a name="sec:source"><TT>op(a, b)</TT></a></TD> |
| <TD>See below for semantics.<br> |
| Return type: <TT>X</TT> |
| </TD> |
| </TR> |
| |
| <tr> |
| <TD><TT>a == b</TT></TD> |
| <TD>Returns true if <tt>a</tt> and <tt>b</tt> represent |
| the same element of <i>S</i>.<br> |
| Return type: <TT>bool</TT> |
| </TD> |
| </TR> |
| |
| <tr> |
| <TD><TT>a != b</TT></TD> |
| <TD>Returns true if <tt>a</tt> and <tt>b</tt> represent |
| different elements of <i>S</i>.<br> |
| Return type: <TT>bool</TT> |
| </TD> |
| </TR> |
| |
| </TABLE> |
| |
| <P> |
| |
| <H3>Invariants</H3> |
| |
| <UL> |
| <li>Closure<br> |
| The result of <tt>op(a, b)</tt> is also an element of <i>S</i>. |
| <li>Associativity<br> |
| <tt>op(op(a, b), c) == op(a, op(b, c))</tt> |
| <li>Definition of Identity Element<br> |
| <tt>op(a, i) == a</tt> |
| </UL> |
| |
| <br> |
| <HR> |
| <TABLE> |
| <TR valign=top> |
| <TD nowrap>Copyright © 2000-2001</TD><TD> |
| <A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, |
| Indiana University (<A |
| HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)<br> |
| <A HREF="http://www.boost.org/people/liequan_lee.htm">Lie-Quan Lee</A>, Indiana University (<A HREF="mailto:llee@cs.indiana.edu">llee@cs.indiana.edu</A>)<br> |
| <A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>, |
| Indiana University (<A |
| HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>) |
| </TD></TR></TABLE> |
| |
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