123 lines
2.9 KiB
HTML
123 lines
2.9 KiB
HTML
<HTML>
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<!--
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Copyright (c) Jeremy Siek, Lie-Quan Lee, and Andrew Lumsdaine 2001
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Monoid</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="concept:Monoid"></A>
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Monoid
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</H1>
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A <i>Monoid</i> is a concept that describes a simple kind of algebraic
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system. A <b>monoid</b> consists of a set of elements <i>S</i>, a
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binary operation, and an identity element. The C++ representation of a
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monoid consists of a function object that implements the binary
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operation, a set of objects that represent the elements of <i>S</i>,
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and an object that represents the identity element.
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<H3>Refinement of</H3>
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The element type must be a model of <a
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href="../../utility/Assignable.html">Assignable</a> and <a
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href="../../utility/CopyConstructible.html">CopyConstructible</a>.
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The function object type must be a model of <a
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href="http://www.boost.org/sgi/stl/BinaryFunction.html">BinaryFunction</a>.
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<h3>Notation</h3>
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<Table>
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<TR>
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<TD><tt>X</tt></TD>
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<TD>Is the element type of the Monoid.</TD>
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</TR>
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<TR>
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<TD><tt>a, b</tt></TD>
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<TD>are objects of type <tt>X</tt>.</TD>
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</TR>
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<TR>
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<TD><tt>op</tt></TD>
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<TD>Is the function object implementing the Monoid operation.</TD>
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</TR>
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<TR>
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<TD><tt>i</tt></TD>
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<TD>is an object of type <tt>X</tt> and is the identity element
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for the Monoid.</TD>
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</TR>
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</table>
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<h3>Valid Expressions</h3>
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<Table border>
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<tr>
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<td><a name="sec:source"><TT>op(a, b)</TT></a></TD>
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<TD>See below for semantics.<br>
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Return type: <TT>X</TT>
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</TD>
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</TR>
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<tr>
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<TD><TT>a == b</TT></TD>
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<TD>Returns true if <tt>a</tt> and <tt>b</tt> represent
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the same element of <i>S</i>.<br>
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Return type: <TT>bool</TT>
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</TD>
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</TR>
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<tr>
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<TD><TT>a != b</TT></TD>
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<TD>Returns true if <tt>a</tt> and <tt>b</tt> represent
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different elements of <i>S</i>.<br>
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Return type: <TT>bool</TT>
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</TD>
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</TR>
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</TABLE>
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<P>
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<H3>Invariants</H3>
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<UL>
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<li>Closure<br>
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The result of <tt>op(a, b)</tt> is also an element of <i>S</i>.
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<li>Associativity<br>
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<tt>op(op(a, b), c) == op(a, op(b, c))</tt>
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<li>Definition of Identity Element<br>
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<tt>op(a, i) == a</tt>
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</UL>
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
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<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>,
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Indiana University (<A
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HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)<br>
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<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>
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<A HREF="https://homes.cs.washington.edu/~al75">Andrew Lumsdaine</A>,
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Indiana University (<A
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HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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