<p>This is the manual for the <strong class="pkg">Semigroups</strong> package for <strong class="pkg">GAP</strong> version 5.5.4. <strong class="pkg">Semigroups</strong> 5.5.4 is a distant descendant of the <span class="URL"><a href=" http://schmidt.nuigalway.ie/monoid/index.html">Monoid package for GAP 3</a></span> by Goetz Pfeiffer, Steve A. Linton, Edmund F. Robertson, and Nik Ruskuc.</p>
<p>From Version 3.0.0, <strong class="pkg">Semigroups</strong> includes a copy of the <span class="URL"><a href="https://libsemigroups.readthedocs.io/en/latest/">libsemigroups</a></span> C++ library which contains implementations of the Froidure-Pin, Todd-Coxeter, and Knuth-Bendix algorithms (among others) that <strong class="pkg">Semigroups</strong> utilises.</p>
<dl>
<dt><strong class="Mark">Part I: elements</strong></dt>
<dd><p>the different types of elements that are introduced in <strong class="pkg">Semigroups</strong> are described in Chapters <a href="chap3_mj.html#X7C18DB427C9C0917"><span class="RefLink">3</span></a>, <a href="chap4_mj.html#X85A717D1790B7BB5"><span class="RefLink">4</span></a>, and <a href="chap5_mj.html#X82D6B7FE7CAC0AFA"><span class="RefLink">5</span></a>. These include <code class="func">Bipartition</code> (<a href="chap3_mj.html#X7E052E6378A5B758"><span class="RefLink">3.2-1</span></a>), <code class="func">PBR</code> (<a href="chap4_mj.html#X82A8646F7C70CF3B"><span class="RefLink">4.2-1</span></a>), and <code class="func">Matrix</code> (<a href="chap5_mj.html#X7DCA234C86ED8BD3"><span class="RefLink">5.1-5</span></a>), which supplement those already defined in the <strong class="pkg">GAP</strong> library, such as <code class="func">Transformation</code> (<a href="../../../doc/ref/chap53_mj.html#X86ADBDE57A20E323"><span class="RefLink">Reference: Transformation for an image list</span></a>) or <code class="func">PartialPerm</code> (<a href="../../../doc/ref/chap54_mj.html#X8538BAE77F2FB2F8"><span class="RefLink">Reference: PartialPerm for a domain and image</span></a>).</p>
</dd>
<dt><strong class="Mark">Part II: semigroups and monoids defined by generating sets</strong></dt>
<dd><p>functions and operations for creating semigroups and monoids defined by generating sets (of the type described in Part I) are described in Chapter <a href="chap6_mj.html#X7995B4F18672DDB0"><span class="RefLink">6</span></a>.</p>
</dd>
<dt><strong class="Mark">Part III: standard examples and constructions</strong></dt>
<dd><p>standard examples of semigroups, such as <code class="func">FullBooleanMatMonoid</code> (<a href="chap7_mj.html#X7B20103D84E010EF"><span class="RefLink">7.6-1</span></a>) or <code class="func">UniformBlockBijectionMonoid</code> (<a href="chap7_mj.html#X8301C61384168D6F"><span class="RefLink">7.3-8</span></a>), are described in Chapter <a href="chap7_mj.html#X7C76D1DC7DAF03D3"><span class="RefLink">7</span></a>, and standard constructions, such as <code class="func">DirectProduct</code> (<a href="chap8_mj.html#X861BA02C7902A4F4"><span class="RefLink">8.1-1</span></a>) are given in Chapter <a href="chap8_mj.html#X86EE8DC987BA646E"><span class="RefLink">8</span></a>.</p>
</dd>
<dt><strong class="Mark">Part IV: the structure of a semigroup or monoid</strong></dt>
<dd><p>the functionality for determining various structural properties of a given semigroup or monoid are described in Chapters <a href="chap9_mj.html#X83629803819C4A6F"><span class="RefLink">9</span></a>, <a href="chap10_mj.html#X80C6C718801855E9"><span class="RefLink">10</span></a>, <a href="chap11_mj.html#X7C75B1DB81C7779B"><span class="RefLink">11</span></a>, and <a href="chap12_mj.html#X78274024827F306D"><span class="RefLink">12</span></a>.</p>
</dd>
<dt><strong class="Mark">Part V: congruences, quotients, and homomorphisms</strong></dt>
<dd><p>methods for creating and manipulating congruences and homomorphisms are described by Chapters <a href="chap13_mj.html#X82BD951079E3C349"><span class="RefLink">13</span></a> and <a href="chap14_mj.html#X861935DB81A478C2"><span class="RefLink">14</span></a>.</p>
</dd>
<dt><strong class="Mark">Part VI: finitely presented semigroups and monoids</strong></dt>
<dd><p>methods for finitely presented semigroups and monoids, in particular, for Tietze transformations can be found in Chapters <a href="chap15_mj.html#X7F11EF307D4F409B"><span class="RefLink">15</span></a>.</p>
</dd>
<dt><strong class="Mark">Part VII: utilities and helper functions</strong></dt>
<dd><p>functions for reading and writing semigroups and their elements, and for visualising semigroups, and some of their elements, can be found in Chapters <a href="chap16_mj.html#X80E82C6785300A86"><span class="RefLink">16</span></a> and <a href="chap17_mj.html#X80CDCB927B3E5BB9"><span class="RefLink">17</span></a>.</p>
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