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<HTML><!-- -- Copyright (c) 1996-1999 -- Silicon Graphics Computer Systems, Inc. -- -- Permission to use, copy, modify, distribute and sell this software -- and its documentation for any purpose is hereby granted without fee, -- provided that the above copyright notice appears in all copies and -- that both that copyright notice and this permission notice appear -- in supporting documentation. Silicon Graphics makes no -- representations about the suitability of this software for any -- purpose. It is provided "as is" without express or implied warranty. -- -- Copyright (c) 1994 -- Hewlett-Packard Company -- -- Permission to use, copy, modify, distribute and sell this software -- and its documentation for any purpose is hereby granted without fee, -- provided that the above copyright notice appears in all copies and -- that both that copyright notice and this permission notice appear -- in supporting documentation. Hewlett-Packard Company makes no -- representations about the suitability of this software for any -- purpose. It is provided "as is" without express or implied warranty. -- --><Head><Title>MonoidOperation</Title><!-- Generated by htmldoc --></HEAD><BODY TEXT="#000000" LINK="#006600" ALINK="#003300" VLINK="#7C7F87" BGCOLOR="#FFFFFF"><A HREF="/"><IMG SRC="/images/common/sgilogo_small.gif" ALT="SGI Logo" WIDTH="80" HEIGHT="72" BORDER="0"></A><P><!--end header--><BR Clear><H1>MonoidOperation</H1><Table CellPadding=0 CellSpacing=0 width=100%><TR><TD Align=left><Img src = "functors.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD><TD Align=right><Img src = "concept.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD></TR><TR><TD Align=left VAlign=top><b>Category</b>: functors</TD><TD Align=right VAlign=top><b>Component type</b>: concept</TD></TR></Table><h3>Description</h3>A Monoid Operation is a special sort of <A href="BinaryFunction.html">Binary Function</A>. A<A href="BinaryFunction.html">Binary Function</A> must satisfy three conditions in order tobe a Monoid Operation. First, its first argument type and secondargument type must be the same, and its result type must be thesame as its argument type. Second, there must be an identityelement. Third, the operation must be associative. Examplesof Monoid Operations are addition and multiplication. <A href="#1">[1]</A><h3>Refinement of</h3><A href="BinaryFunction.html">Binary Function</A><h3>Associated types</h3><Table border><TR><TD VAlign=top>Argument type</TD><TD VAlign=top>The type of the Monoid Operation's first argument and second argument, and also the type returned when the Monoid Operation is returned.</TD></tr></table><h3>Notation</h3><Table><TR><TD VAlign=top><tt>F</tt></TD><TD VAlign=top>A type that is a model of MonoidOperation</TD></TR><TR><TD VAlign=top><tt>T</tt></TD><TD VAlign=top><tt>F</tt>'s argument type.</TD></TR><TR><TD VAlign=top><tt>f</tt></TD><TD VAlign=top>Object of type <tt>F</tt></TD></TR><TR><TD VAlign=top><tt>x</tt>, <tt>y</tt>, <tt>z</tt></TD><TD VAlign=top>Objects of type <tt>T</tt></TD></tr></table><h3>Definitions</h3>A type <tt>F</tt> that is a model of <A href="BinaryFunction.html">binary function</A> is <i>associative</i> if<tt>F</tt>'s first argument type, second argument type, and result type arethe same, and if, for every object <tt>f</tt> of type <tt>F</tt> and for every objects <tt>x</tt>, <tt>y</tt>, and <tt>z</tt> of <tt>F</tt>'s argument type, <tt>f(x, f(y, z))</tt> is the same as <tt>f(f(x, y), z)</tt>. <A href="#2">[2]</A><h3>Valid Expressions</h3>In addition to the expressions described in the <A href="BinaryFunction.html">Binary Function</A>requirements, the following expressions must be valid.<Table border><TR><TH>Name</TH><TH>Expression</TH><TH>Type requirements</TH><TH>Return type</TH></TR><TR><TD VAlign=top>Function call</TD><TD VAlign=top><tt>f(x, y)</tt></TD><TD VAlign=top> </TD><TD VAlign=top><tt>T</tt></TD></TR><TR><TD VAlign=top>Identity element</TD><TD VAlign=top><tt>identity_element(f)</tt> <A href="#3">[3]</A></TD><TD VAlign=top> </TD><TD VAlign=top><tt>T</tt></TD></tr></table><h3>Expression semantics</h3><Table border><TR><TH>Name</TH><TH>Expression</TH><TH>Precondition</TH><TH>Semantics</TH><TH>Postcondition</TH></TR><TR><TD VAlign=top>Function call</TD><TD VAlign=top><tt>f(x, y)</tt></TD><TD VAlign=top><tt>x</tt> and <tt>y</tt> are in the domain of <tt>f</tt>.</TD><TD VAlign=top>Calls <tt>f</tt> with <tt>x</tt> and <tt>y</tt> as arguments.</TD><TD VAlign=top> </TD></TR><TR><TD VAlign=top>Identity element</TD><TD VAlign=top><tt>identity_element(f)</tt></TD><TD VAlign=top> </TD><TD VAlign=top>Returns the monoid's identity element. That is, the return value is a value <tt>id</tt> of type <tt>T</tt> such that, for all <tt>x</tt> in the domain of <tt>f</tt>, <tt>f(x, id)</tt> and <tt>f(id, x)</tt> both return <tt>x</tt>.</TD><TD VAlign=top> </TD></tr></table><h3>Complexity guarantees</h3><h3>Invariants</h3><Table border><TR><TD VAlign=top>Associativity</TD><TD VAlign=top>For any <tt>x</tt>, <tt>y</tt>, and <tt>z</tt> of type <tt>T</tt>, <tt>f(x, f(y, z))</tt> and <tt>f(f(x, y), z)</tt> return the same value. <A href="#4">[4]</A></TD></TR><TR><TD VAlign=top>Identity element.</TD><TD VAlign=top>There exists some element <tt>id</tt> of type <tt>T</tt> such that, for all <tt>x</tt> of type <tt>T</tt>, <tt>f(x, id)</tt> and <tt>f(id, x)</tt> both return <tt>x</tt>. The expression <tt>identity_element(f)</tt> returns <tt>id</tt>.</TD></tr></table><h3>Models</h3><UL><LI><tt><A href="plus.html">plus</A><int></tt><LI><tt><A href="times.html">multiplies</A><double></tt></UL><h3>Notes</h3><P><A name="1">[1]</A>A monoid is one of three closely related algebraic structures.A <i>semigroup</i> is a set S, and a binary operation *, with theproperties that * is closed on S (that is, if x and y are elements ofS then x * y is also a member of S) and that * is associative (thatis, if x, y, and z are elements of S, then x * (y * z) = (x * y) * z).A <i>monoid</i> is a semigroup that has an identity element. That is,there exists some element id such that, for all x in S, x * id = id * x =x. Finally, a <i>group</i> is a monoid with the property that every elementhas an inverse. That is, for every x in S, there exists an elementxi such that x * xi = xi * x = id. As an example, the set of real numbers under multiplication is a monoid (the identity elementis 1), but it isn't a group. It isn't a group because 0 has no inverse.<P><A name="2">[2]</A>Mathematics textbooks typically write this as an equation,instead of using words like "is the same as". We can't useequality in this definition, however, because <tt>F</tt>'s argument typemight not be <A href="EqualityComparable.html">equality comparable</A>. If <tt>F</tt>'s argument type is<A href="EqualityComparable.html">equality comparable</A>, however, then these two expression are expectedto be equal: the condition of associativity becomes<tt>f(x, f(y, z)) == f(f(x, y), z)</tt><P><A name="3">[3]</A>This is implemented as an overloaded function. The function<tt>identity_element</tt> is defined, in the standard header <A href="functional">functional</A>, and the nonstandard backward-compatibility header <A href="function.h">function.h</A>,for arguments of type <tt><A href="plus.html">plus</A><T></tt> and <tt><A href="times.html">multiplies</A><T></tt>. If you define a new Monoid Operation <tt>F</tt> (matrix multiplication, forexample), you must overload <tt>identity_element</tt> for arguments of type<tt>F</tt>. The <tt>identity_element</tt> function is an SGI extension; it is notpart of the C++ standard. <P><A name="4">[4]</A>Associativity is not the same as commutativity. That is, the requirementthat <tt>x * (y * z) == (x * y) * z</tt> is completely unrelated tothe requirement that <tt>x * y == y * x</tt>. Monoid operations arerequired to be associative, but they are not required to be commutative. As an example, square matrices under multiplicationform a monoid even though matrix multiplication is not commutative.<h3>See also</h3><A href="BinaryFunction.html">Binary Function</A>, <tt><A href="plus.html">plus</A></tt>, <tt><A href="times.html">multiplies</A></tt><!-- start footer --><!-- Footer Begins --><STYLE TYPE="text/css"><!--TD.footer, TD.footer A{ font-family: Arial, helvetica, sans-serif; font-size: 8pt;}A.home {font-family: Arial, helvetica, sans-serif;}--></STYLE><P><A CLASS="home" HREF="index.html">STL Home</A><P><TABLE WIDTH="600" CELLPADDING="0" CELLPADDING="0" BORDER="0"> <TR> <TD ALIGN="RIGHT" CLASS="footer"><A HREF="/company_info/terms.html" TARGET="_top">terms of use</A> | <A HREF="/company_info/privacy.html" TARGET="_top">privacy policy</A></TD> <TD ALIGN="CENTER" CLASS="footer"> | </TD> <TD ALIGN="LEFT" CLASS="footer"><A HREF="/cgi-bin/feedback/" TARGET="_top">contact us</A></TD> </TR><TR> <TD ALIGN="RIGHT" CLASS="footer">Copyright © 1993-2003 Silicon Graphics, Inc. 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