1<HTML> 2<!-- 3 Copyright (c) Jeremy Siek, Lie-Quan Lee, and Andrew Lumsdaine 2001 4 5 Distributed under the Boost Software License, Version 1.0. 6 (See accompanying file LICENSE_1_0.txt or copy at 7 http://www.boost.org/LICENSE_1_0.txt) 8 --> 9<Head> 10<Title>Monoid</Title> 11<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" 12 ALINK="#ff0000"> 13<IMG SRC="../../../boost.png" 14 ALT="C++ Boost" width="277" height="86"> 15 16<BR Clear> 17 18<H1><A NAME="concept:Monoid"></A> 19Monoid 20</H1> 21 22A <i>Monoid</i> is a concept that describes a simple kind of algebraic 23system. A <b>monoid</b> consists of a set of elements <i>S</i>, a 24binary operation, and an identity element. The C++ representation of a 25monoid consists of a function object that implements the binary 26operation, a set of objects that represent the elements of <i>S</i>, 27and an object that represents the identity element. 28 29 30<H3>Refinement of</H3> 31 32The element type must be a model of <a 33href="../../utility/Assignable.html">Assignable</a> and <a 34href="../../utility/CopyConstructible.html">CopyConstructible</a>. 35The function object type must be a model of <a 36href="http://www.boost.org/sgi/stl/BinaryFunction.html">BinaryFunction</a>. 37 38<h3>Notation</h3> 39 40<Table> 41<TR> 42<TD><tt>X</tt></TD> 43<TD>Is the element type of the Monoid.</TD> 44</TR> 45 46<TR> 47<TD><tt>a, b</tt></TD> 48<TD>are objects of type <tt>X</tt>.</TD> 49</TR> 50 51<TR> 52<TD><tt>op</tt></TD> 53<TD>Is the function object implementing the Monoid operation.</TD> 54</TR> 55 56<TR> 57<TD><tt>i</tt></TD> 58<TD>is an object of type <tt>X</tt> and is the identity element 59for the Monoid.</TD> 60</TR> 61 62 63</table> 64 65<h3>Valid Expressions</h3> 66 67<Table border> 68 69<tr> 70<td><a name="sec:source"><TT>op(a, b)</TT></a></TD> 71<TD>See below for semantics.<br> 72Return type: <TT>X</TT> 73</TD> 74</TR> 75 76<tr> 77<TD><TT>a == b</TT></TD> 78<TD>Returns true if <tt>a</tt> and <tt>b</tt> represent 79 the same element of <i>S</i>.<br> 80Return type: <TT>bool</TT> 81</TD> 82</TR> 83 84<tr> 85<TD><TT>a != b</TT></TD> 86<TD>Returns true if <tt>a</tt> and <tt>b</tt> represent 87 different elements of <i>S</i>.<br> 88Return type: <TT>bool</TT> 89</TD> 90</TR> 91 92</TABLE> 93 94<P> 95 96<H3>Invariants</H3> 97 98<UL> 99 <li>Closure<br> 100 The result of <tt>op(a, b)</tt> is also an element of <i>S</i>. 101 <li>Associativity<br> 102 <tt>op(op(a, b), c) == op(a, op(b, c))</tt> 103 <li>Definition of Identity Element<br> 104 <tt>op(a, i) == a</tt> 105</UL> 106 107<br> 108<HR> 109<TABLE> 110<TR valign=top> 111<TD nowrap>Copyright © 2000-2001</TD><TD> 112<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, 113Indiana University (<A 114HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)<br> 115<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> 116<A HREF="https://homes.cs.washington.edu/~al75">Andrew Lumsdaine</A>, 117Indiana University (<A 118HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>) 119</TD></TR></TABLE> 120 121</BODY> 122</HTML> 123
