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