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<div class="header">
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<a href="#var-members">Variables</a>  </div>
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<div class="title">Monoid<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
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<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
<p>The <code>Monoid</code> concept represents data types with an associative binary operation that has an identity. </p>
<p>Specifically, a <a href="http://en.wikipedia.org/wiki/Monoid">Monoid</a> is a basic algebraic structure typically used in mathematics to construct more complex algebraic structures like <code>Group</code>s, <code>Ring</code>s and so on. They are useful in several contexts, notably to define the properties of numbers in a granular way. At its core, a <code>Monoid</code> is a set <code>S</code> of objects along with a binary operation (let's say <code>+</code>) that is associative and that has an identity in <code>S</code>. There are many examples of <code>Monoid</code>s:</p><ul>
<li>strings with concatenation and the empty string as the identity</li>
<li>integers with addition and <code>0</code> as the identity</li>
<li>integers with multiplication and <code>1</code> as the identity</li>
<li>many others...</li>
</ul>
<p>As you can see with the integers, there are some sets that can be viewed as a monoid in more than one way, depending on the choice of the binary operation and identity. The method names used here refer to the monoid of integers under addition; <code>plus</code> is the binary operation and <code>zero</code> is the identity element of that operation.</p>
<h2>Minimal complete definition </h2>
<p><code>plus</code> and <code>zero</code> satisfying the laws</p>
<h2>Laws </h2>
<p>For all objects <code>x</code>, <code>y</code> and <code>z</code> of a <code>Monoid</code> <code>M</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>(zero&lt;M&gt;(), x) == x                    <span class="comment">// left zero</span></div><div class="line"><a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, zero&lt;M&gt;()) == x                    <span class="comment">// right zero</span></div><div class="line"><a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(y, z)) == <a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(<a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, y), z) <span class="comment">// associativity</span></div></div><!-- fragment --><h2>Concrete models </h2>
<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>
<h2>Free model for non-boolean arithmetic data types </h2>
<p>A data type <code>T</code> is arithmetic if <code>std::is_arithmetic&lt;T&gt;::value</code> is true. For a non-boolean arithmetic data type <code>T</code>, a model of <code>Monoid</code> is automatically defined by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, y) = (x + y)</div><div class="line">zero&lt;T&gt;() = static_cast&lt;T&gt;(0)</div></div><!-- fragment --><blockquote class="doxtable">
<h4>Rationale for not making <code>bool</code> a <code>Monoid</code> by default</h4>
<p>First, it makes no sense whatsoever to define an additive <code>Monoid</code> over the <code>bool</code> type. Also, it could make sense to define a <code>Monoid</code> with logical conjunction or disjunction. However, C++ allows <code>bool</code>s to be added, and the method names of this concept really suggest addition. In line with the principle of least surprise, no model is provided by default. </p>
</blockquote>
<h2>Structure-preserving functions </h2>
<p>Let <code>A</code> and <code>B</code> be two <code>Monoid</code>s. A function <code>f : A -&gt; B</code> is said to be a <a href="http://en.wikipedia.org/wiki/Monoid#Monoid_homomorphisms">Monoid morphism</a> if it preserves the monoidal 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><div class="line">f(zero&lt;A&gt;()) == zero&lt;B&gt;()</div></div><!-- fragment --><p> Functions with these properties interact nicely with <code>Monoid</code>s, which is why they are given such a special treatment. </p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="var-members"></a>
Variables</h2></td></tr>
<tr class="memitem:gaeb5d4a1e967e319712f9e4791948896c"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">boost::hana::plus</a></td></tr>
<tr class="memdesc:gaeb5d4a1e967e319712f9e4791948896c"><td class="mdescLeft">&#160;</td><td class="mdescRight">Associative binary operation on a <code>Monoid</code>.  <a href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">More...</a><br /></td></tr>
<tr class="separator:gaeb5d4a1e967e319712f9e4791948896c"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:gad459ac17b6bab8ead1cae7de0032f3c6"><td class="memTemplParams" colspan="2">template&lt;typename M &gt; </td></tr>
<tr class="memitem:gad459ac17b6bab8ead1cae7de0032f3c6"><td class="memTemplItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__group-Monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">boost::hana::zero</a></td></tr>
<tr class="memdesc:gad459ac17b6bab8ead1cae7de0032f3c6"><td class="mdescLeft">&#160;</td><td class="mdescRight">Identity of <code>plus</code>.  <a href="group__group-Monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">More...</a><br /></td></tr>
<tr class="separator:gad459ac17b6bab8ead1cae7de0032f3c6"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<h2 class="groupheader">Variable Documentation</h2>
<a id="gaeb5d4a1e967e319712f9e4791948896c"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gaeb5d4a1e967e319712f9e4791948896c">&#9670;&nbsp;</a></span>plus</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2plus_8hpp.html">boost/hana/fwd/plus.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Associative binary operation on a <code>Monoid</code>. </p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x,y</td><td>Two objects to combine with the <code>Monoid</code>'s binary operation.</td></tr>
  </table>
  </dd>
</dl>
<h2>Cross-type version of the method </h2>
<p>The <code>plus</code> method is "overloaded" to handle distinct data types with certain properties. Specifically, <code>plus</code> is defined for <em>distinct</em> data types <code>A</code> and <code>B</code> such that</p><ol type="1">
<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>
<li><code>A</code>, <code>B</code> and <code>C</code> are all <code>Monoid</code>s when taken individually</li>
<li><code>to&lt;C&gt; : A -&gt; B</code> and <code>to&lt;C&gt; : B -&gt; C</code> are <code>Monoid</code>-embeddings, as determined by the <code>is_embedding</code> metafunction.</li>
</ol>
<p>The definition of <code>plus</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-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, y) = <a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div></div><!-- fragment --><h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="plus_8hpp.html">boost/hana/plus.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">hana::plus</a>(hana::int_c&lt;3&gt;, hana::int_c&lt;5&gt;) == hana::int_c&lt;8&gt;);</div><div class="line">static_assert(<a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">hana::plus</a>(1, 2) == 3, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">static_assert(<a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">hana::plus</a>(1.5f, 2.4) == 3.9, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<a id="gad459ac17b6bab8ead1cae7de0032f3c6"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gad459ac17b6bab8ead1cae7de0032f3c6">&#9670;&nbsp;</a></span>zero</h2>

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template&lt;typename M &gt; </div>
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<p><code>#include &lt;<a class="el" href="fwd_2zero_8hpp.html">boost/hana/fwd/zero.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= []() -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Identity of <code>plus</code>. </p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
  <table class="tparams">
    <tr><td class="paramname">M</td><td>The tag (a <code>Monoid</code>) of the returned identity.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="zero_8hpp.html">boost/hana/zero.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">hana::zero</a>&lt;hana::integral_constant_tag&lt;int&gt;&gt;() == hana::int_c&lt;0&gt;);</div><div class="line">static_assert(hana::zero&lt;long&gt;() == 0l, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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