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<a href="#var-members">Variables</a> </div>
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<div class="headertitle">
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<div class="title">Group<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div> </div>
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</div><!--header-->
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<div class="contents">
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<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
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<p>The <code>Group</code> concept represents <code>Monoid</code>s where all objects have an inverse w.r.t. the <code>Monoid</code>'s binary operation. </p>
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<p>A <a href="http://en.wikipedia.org/wiki/Group_(mathematics)">Group</a> is an algebraic structure built on top of a <code>Monoid</code> which adds the ability to invert the action of the <code>Monoid</code>'s binary operation on any element of the set. Specifically, a <code>Group</code> is a <code>Monoid</code> <code>(S, +)</code> such that every element <code>s</code> in <code>S</code> has an inverse (say ‘s’`) which is such that </p><div class="fragment"><div class="line">s + s<span class="stringliteral">' == s'</span> + s == identity of the Monoid</div>
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</div><!-- fragment --><p>There are many examples of <code>Group</code>s, one of which would be the additive <code>Monoid</code> on integers, where the inverse of any integer <code>n</code> is the integer <code>-n</code>. The method names used here refer to exactly this model.</p>
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<h2><a class="anchor" id="autotoc_md111"></a>
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Minimal complete definitions</h2>
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<ol type="1">
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<li><code>minus</code><br />
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When <code>minus</code> is specified, the <code>negate</code> method is defaulted by setting <div class="fragment"><div class="line"><a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x) = <a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(zero<G>(), x)</div>
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</div><!-- fragment --></li>
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<li><code>negate</code><br />
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When <code>negate</code> is specified, the <code>minus</code> method is defaulted by setting <div class="fragment"><div class="line"><a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) = <a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(y))</div>
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</div><!-- fragment --></li>
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</ol>
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<h2><a class="anchor" id="autotoc_md112"></a>
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Laws</h2>
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<p>For all objects <code>x</code> of a <code>Group</code> <code>G</code>, the following laws must be satisfied: </p><div class="fragment"><div class="line"><a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x)) == zero<G>() <span class="comment">// right inverse</span></div>
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<div class="line"><a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(<a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x), x) == <a class="code" href="group__group-_monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">zero</a><G>() <span class="comment">// left inverse</span></div>
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</div><!-- fragment --><h2><a class="anchor" id="autotoc_md113"></a>
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Refined concept</h2>
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<p><code>Monoid</code></p>
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<h2><a class="anchor" id="autotoc_md114"></a>
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Concrete models</h2>
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<p><code><a class="el" href="structboost_1_1hana_1_1integral__constant.html" title="Compile-time value of an integral type.">hana::integral_constant</a></code></p>
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<h2><a class="anchor" id="autotoc_md115"></a>
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Free model for non-boolean arithmetic data types</h2>
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<p>A data type <code>T</code> is arithmetic if <code>std::is_arithmetic<T>::value</code> is true. For a non-boolean arithmetic data type <code>T</code>, a model of <code>Group</code> is automatically defined by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) = (x - y)</div>
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<div class="line"><a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x) = -x</div>
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</div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>The rationale for not providing a Group model for <code>bool</code> is the same as for not providing a <code>Monoid</code> model.</dd></dl>
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<h2><a class="anchor" id="autotoc_md116"></a>
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Structure-preserving functions</h2>
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<p>Let <code>A</code> and <code>B</code> be two <code>Group</code>s. A function <code>f : A -> B</code> is said to be a <a href="http://en.wikipedia.org/wiki/Group_homomorphism">Group morphism</a> if it preserves the group structure between <code>A</code> and <code>B</code>. Rigorously, for all objects <code>x, y</code> of data type <code>A</code>, </p><div class="fragment"><div class="line">f(<a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, y)) == <a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(f(x), f(y))</div>
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</div><!-- fragment --><p> Because of the <code>Group</code> structure, it is easy to prove that the following will then also be satisfied: </p><div class="fragment"><div class="line">f(<a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x)) == <a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(f(x))</div>
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<div class="line">f(zero<A>()) == zero<B>()</div>
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</div><!-- fragment --><p> Functions with these properties interact nicely with <code>Group</code>s, which is why they are given such a special treatment. </p>
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<table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="var-members"></a>
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Variables</h2></td></tr>
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<tr class="memitem:ga2020c526324f361a2b990fe8d1b07c20"><td class="memItemLeft" align="right" valign="top">constexpr auto </td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">boost::hana::minus</a></td></tr>
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<tr class="memdesc:ga2020c526324f361a2b990fe8d1b07c20"><td class="mdescLeft"> </td><td class="mdescRight">Subtract two elements of a group. <a href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">More...</a><br /></td></tr>
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<tr class="separator:ga2020c526324f361a2b990fe8d1b07c20"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ga02e81002f40ba52eac4cf1974c7e0cdb"><td class="memItemLeft" align="right" valign="top">constexpr auto </td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">boost::hana::negate</a></td></tr>
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<tr class="memdesc:ga02e81002f40ba52eac4cf1974c7e0cdb"><td class="mdescLeft"> </td><td class="mdescRight">Return the inverse of an element of a group. <a href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">More...</a><br /></td></tr>
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<tr class="separator:ga02e81002f40ba52eac4cf1974c7e0cdb"><td class="memSeparator" colspan="2"> </td></tr>
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</table>
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<h2 class="groupheader">Variable Documentation</h2>
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<a id="ga2020c526324f361a2b990fe8d1b07c20"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ga2020c526324f361a2b990fe8d1b07c20">◆ </a></span>minus</h2>
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<td class="memname">constexpr auto boost::hana::minus</td>
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<span class="mlabels"><span class="mlabel">constexpr</span></span> </td>
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<p><code>#include <<a class="el" href="fwd_2minus_8hpp.html">boost/hana/fwd/minus.hpp</a>></code></p>
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<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&& x, <span class="keyword">auto</span>&& y) -> decltype(<span class="keyword">auto</span>) {</div>
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<div class="line"> <span class="keywordflow">return</span> tag-dispatched;</div>
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<div class="line"> }</div>
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</div><!-- fragment -->
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<p>Subtract two elements of a group. </p>
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<p>Specifically, this performs the <code>Monoid</code> operation on the first argument and on the inverse of the second argument, thus being equivalent to: </p><div class="fragment"><div class="line"><a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) == <a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(y))</div>
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</div><!-- fragment --><h2><a class="anchor" id="autotoc_md287"></a>
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Cross-type version of the method</h2>
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<p>The <code>minus</code> method is "overloaded" to handle distinct data types with certain properties. Specifically, <code>minus</code> is defined for <em>distinct</em> data types <code>A</code> and <code>B</code> such that</p><ol type="1">
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<li><code>A</code> and <code>B</code> share a common data type <code>C</code>, as determined by the <code>common</code> metafunction</li>
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<li><code>A</code>, <code>B</code> and <code>C</code> are all <code>Group</code>s when taken individually</li>
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<li><code>to<C> : A -> B</code> and <code>to<C> : B -> C</code> are <code>Group</code>-embeddings, as determined by the <code>is_embedding</code> metafunction.</li>
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</ol>
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<p>The definition of <code>minus</code> for data types satisfying the above properties is obtained by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) = <a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(to<C>(x), to<C>(y))</div>
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</div><!-- fragment --><h2><a class="anchor" id="autotoc_md288"></a>
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Example</h2>
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<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div>
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<div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div>
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<div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div>
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<div class="line"> </div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>></span></div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>></span></div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>></span></div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="minus_8hpp.html">boost/hana/minus.hpp</a>></span></div>
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<div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div>
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<div class="line"> </div>
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<div class="line"> </div>
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<div class="line"><span class="keywordtype">int</span> main() {</div>
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<div class="line"> <a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">hana::minus</a>(hana::int_c<3>, hana::int_c<5>) == hana::int_c<-2>);</div>
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<div class="line"> static_assert(<a class="code" href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">hana::minus</a>(1, 2) == -1, <span class="stringliteral">""</span>);</div>
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<div class="line">}</div>
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</div><!-- fragment -->
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<a id="ga02e81002f40ba52eac4cf1974c7e0cdb"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ga02e81002f40ba52eac4cf1974c7e0cdb">◆ </a></span>negate</h2>
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<td class="memname">constexpr auto boost::hana::negate</td>
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<span class="mlabels"><span class="mlabel">constexpr</span></span> </td>
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<p><code>#include <<a class="el" href="fwd_2negate_8hpp.html">boost/hana/fwd/negate.hpp</a>></code></p>
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<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&& x) -> decltype(<span class="keyword">auto</span>) {</div>
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<div class="line"> <span class="keywordflow">return</span> tag-dispatched;</div>
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<div class="line"> }</div>
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</div><!-- fragment -->
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<p>Return the inverse of an element of a group. </p>
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<h2><a class="anchor" id="autotoc_md296"></a>
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Example</h2>
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<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div>
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<div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div>
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<div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div>
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<div class="line"> </div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>></span></div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>></span></div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>></span></div>
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<div class="line"><span class="preprocessor">#include <<a class="code" href="negate_8hpp.html">boost/hana/negate.hpp</a>></span></div>
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<div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div>
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<div class="line"> </div>
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<div class="line"> </div>
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<div class="line"><span class="keywordtype">int</span> main() {</div>
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<div class="line"> <a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">hana::negate</a>(hana::int_c<3>) == hana::int_c<-3>);</div>
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<div class="line"> static_assert(<a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">hana::negate</a>(2) == -2, <span class="stringliteral">""</span>);</div>
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<div class="line">}</div>
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</div><!-- fragment -->
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</div><!-- contents -->
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</div><!-- doc-content -->
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<div class="ttc" id="aequal_8hpp_html"><div class="ttname"><a href="equal_8hpp.html">equal.hpp</a></div><div class="ttdoc">Defines boost::hana::equal.</div></div>
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<div class="ttc" id="agroup__group-assertions_html_ga2c5006540936d9f8880e3a39f4fcc035"><div class="ttname"><a href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a></div><div class="ttdeci">#define BOOST_HANA_CONSTANT_CHECK(...)</div><div class="ttdoc">Equivalent to BOOST_HANA_CONSTANT_ASSERT, but not influenced by the BOOST_HANA_CONFIG_DISABLE_ASSERTI...</div><div class="ttdef"><b>Definition:</b> assert.hpp:239</div></div>
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<div class="ttc" id="agroup__group-_group_html_ga02e81002f40ba52eac4cf1974c7e0cdb"><div class="ttname"><a href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">boost::hana::negate</a></div><div class="ttdeci">constexpr auto negate</div><div class="ttdoc">Return the inverse of an element of a group.</div><div class="ttdef"><b>Definition:</b> negate.hpp:26</div></div>
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<div class="ttc" id="agroup__group-_group_html_ga2020c526324f361a2b990fe8d1b07c20"><div class="ttname"><a href="group__group-_group.html#ga2020c526324f361a2b990fe8d1b07c20">boost::hana::minus</a></div><div class="ttdeci">constexpr auto minus</div><div class="ttdoc">Subtract two elements of a group.</div><div class="ttdef"><b>Definition:</b> minus.hpp:51</div></div>
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<div class="ttc" id="aintegral__constant_8hpp_html"><div class="ttname"><a href="integral__constant_8hpp.html">integral_constant.hpp</a></div><div class="ttdoc">Defines boost::hana::integral_constant.</div></div>
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<div class="ttc" id="agroup__group-_monoid_html_gad459ac17b6bab8ead1cae7de0032f3c6"><div class="ttname"><a href="group__group-_monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">boost::hana::zero</a></div><div class="ttdeci">constexpr auto zero</div><div class="ttdoc">Identity of plus.</div><div class="ttdef"><b>Definition:</b> zero.hpp:30</div></div>
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<div class="ttc" id="agroup__group-_monoid_html_gaeb5d4a1e967e319712f9e4791948896c"><div class="ttname"><a href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">boost::hana::plus</a></div><div class="ttdeci">constexpr auto plus</div><div class="ttdoc">Associative binary operation on a Monoid.</div><div class="ttdef"><b>Definition:</b> plus.hpp:47</div></div>
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<div class="ttc" id="anegate_8hpp_html"><div class="ttname"><a href="negate_8hpp.html">negate.hpp</a></div><div class="ttdoc">Defines boost::hana::negate.</div></div>
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<div class="ttc" id="aminus_8hpp_html"><div class="ttname"><a href="minus_8hpp.html">minus.hpp</a></div><div class="ttdoc">Defines boost::hana::minus.</div></div>
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<div class="ttc" id="anamespaceboost_1_1hana_html"><div class="ttname"><a href="namespaceboost_1_1hana.html">boost::hana</a></div><div class="ttdoc">Namespace containing everything in the library.</div><div class="ttdef"><b>Definition:</b> accessors.hpp:20</div></div>
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<div class="ttc" id="aassert_8hpp_html"><div class="ttname"><a href="assert_8hpp.html">assert.hpp</a></div><div class="ttdoc">Defines macros to perform different kinds of assertions.</div></div>
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