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| 3 | -- Copyright (c) Jeremy Siek, Lie-Quan Lee, and Andrew Lumsdaine 2001 |
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| 5 | -- Permission to use, copy, modify, distribute and sell this software |
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| 9 | -- in supporting documentation. We make no |
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| 11 | -- purpose. It is provided "as is" without express or implied warranty. |
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| 12 | --> |
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| 13 | <Head> |
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| 14 | <Title>Monoid</Title> |
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| 15 | <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" |
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| 16 | ALINK="#ff0000"> |
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| 17 | <IMG SRC="../../../boost.png" |
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| 18 | ALT="C++ Boost" width="277" height="86"> |
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| 19 | |
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| 20 | <BR Clear> |
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| 21 | |
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| 22 | <H1><A NAME="concept:Monoid"></A> |
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| 23 | Monoid |
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| 24 | </H1> |
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| 25 | |
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| 26 | A <i>Monoid</i> is a concept that describes a simple kind of algebraic |
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| 27 | system. A <b>monoid</b> consists of a set of elements <i>S</i>, a |
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| 28 | binary operation, and an identity element. The C++ representation of a |
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| 29 | monoid consists of a function object that implements the binary |
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| 30 | operation, a set of objects that represent the elements of <i>S</i>, |
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| 31 | and an object that represents the identity element. |
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| 32 | |
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| 33 | |
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| 34 | <H3>Refinement of</H3> |
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| 35 | |
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| 36 | The element type must be a model of <a |
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| 37 | href="../../utility/Assignable.html">Assignable</a> and <a |
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| 38 | href="../../utility/CopyConstructible.html">CopyConstructible</a>. |
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| 39 | The function object type must be a model of <a |
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| 40 | href="http://www.sgi.com/tech/stl/BinaryFunction.html">BinaryFunction</a>. |
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| 41 | |
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| 42 | <h3>Notation</h3> |
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| 43 | |
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| 44 | <Table> |
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| 45 | <TR> |
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| 46 | <TD><tt>X</tt></TD> |
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| 47 | <TD>Is the element type of the Monoid.</TD> |
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| 48 | </TR> |
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| 49 | |
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| 50 | <TR> |
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| 51 | <TD><tt>a, b</tt></TD> |
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| 52 | <TD>are objects of type <tt>X</tt>.</TD> |
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| 53 | </TR> |
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| 54 | |
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| 55 | <TR> |
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| 56 | <TD><tt>op</tt></TD> |
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| 57 | <TD>Is the function object implementing the Monoid operation.</TD> |
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| 58 | </TR> |
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| 59 | |
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| 60 | <TR> |
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| 61 | <TD><tt>i</tt></TD> |
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| 62 | <TD>is an object of type <tt>X</tt> and is the identity element |
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| 63 | for the Monoid.</TD> |
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| 64 | </TR> |
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| 65 | |
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| 66 | |
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| 67 | </table> |
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| 68 | |
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| 69 | <h3>Valid Expressions</h3> |
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| 70 | |
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| 71 | <Table border> |
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| 72 | |
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| 73 | <tr> |
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| 74 | <td><a name="sec:source"><TT>op(a, b)</TT></a></TD> |
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| 75 | <TD>See below for semantics.<br> |
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| 76 | Return type: <TT>X</TT> |
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| 77 | </TD> |
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| 78 | </TR> |
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| 79 | |
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| 80 | <tr> |
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| 81 | <TD><TT>a == b</TT></TD> |
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| 82 | <TD>Returns true if <tt>a</tt> and <tt>b</tt> represent |
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| 83 | the same element of <i>S</i>.<br> |
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| 84 | Return type: <TT>bool</TT> |
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| 85 | </TD> |
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| 86 | </TR> |
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| 87 | |
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| 88 | <tr> |
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| 89 | <TD><TT>a != b</TT></TD> |
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| 90 | <TD>Returns true if <tt>a</tt> and <tt>b</tt> represent |
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| 91 | different elements of <i>S</i>.<br> |
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| 92 | Return type: <TT>bool</TT> |
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| 93 | </TD> |
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| 94 | </TR> |
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| 95 | |
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| 96 | </TABLE> |
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| 97 | |
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| 98 | <P> |
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| 99 | |
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| 100 | <H3>Invariants</H3> |
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| 101 | |
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| 102 | <UL> |
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| 103 | <li>Closure<br> |
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| 104 | The result of <tt>op(a, b)</tt> is also an element of <i>S</i>. |
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| 105 | <li>Associativity<br> |
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| 106 | <tt>op(op(a, b), c) == op(a, op(b, c))</tt> |
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| 107 | <li>Definition of Identity Element<br> |
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| 108 | <tt>op(a, i) == a</tt> |
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| 109 | </UL> |
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| 110 | |
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| 111 | <br> |
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| 112 | <HR> |
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| 113 | <TABLE> |
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| 114 | <TR valign=top> |
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| 115 | <TD nowrap>Copyright © 2000-2001</TD><TD> |
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| 116 | <A HREF="../../../people/jeremy_siek.htm">Jeremy Siek</A>, |
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| 117 | Indiana University (<A |
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| 118 | HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)<br> |
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| 119 | <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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| 120 | <A HREF=http://www.osl.iu.edu/~lums>Andrew Lumsdaine</A>, |
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| 121 | Indiana University (<A |
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| 122 | HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>) |
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| 123 | </TD></TR></TABLE> |
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| 124 | |
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| 125 | </BODY> |
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| 126 | </HTML> |
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