<p>We give some examples of semigroups to be used later. We also describe some basic functions that are not directly available from <strong class="pkg">GAP</strong>, but are useful for the purposes of this package.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInverseSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Tests whether a finite semigroup <var class="Arg">S </var>is inverse. It is well-known that it suffices to test whether the idempotents of <var class="Arg">S </var>commute and <var class="Arg">S </var>is regular. The function <code class="code">IsRegularSemigroup </code>is part of <strong class="pkg">GAP</strong>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialTransformation</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A partial transformation is a partial function of a set of integers of the form <span class="SimpleMath">\(\{1,\dots, n\}\)</span>. It is given by means of the list of images <var class="Arg">L</var>. When an element has no image, we write 0. Returns a full transformation on a set with one more element that acts like a zero.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReduceNumberOfGenerators</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a subset <var class="Arg">L</var> of the generators of a semigroup, returns a list of generators of the same semigroup but possibly with less elements than <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupFactorization</code>( <var class="Arg">S</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">L</var> is an element (or list of elements) of the semigroup <var class="Arg">S</var>. Returns a minimal factorization on the generators of <var class="Arg">S</var> of the element(s) of <var class="Arg">L</var>. Works only for transformation semigroups.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GrahamBlocks</code>( <var class="Arg">mat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">mat</var> is a matrix as displayed by <code class="code">DisplayEggBoxOfDClass(D);</code> of a regular D-class <code class="code">D</code>. This function outputs a list <code class="code">[gmat, phi]</code> where <code class="code">gmat</code> is <var class="Arg">mat</var> in Graham's blocks form and phi maps H-classes of gmat to the corresponding ones of mat, i.e., phi[i][j] = [i',j'] where mat[i'][j'] = gmat[i][j]. If the argument to this function is not a matrix corresponding to a regular D-class, the function may abort in error.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightCayleyGraphAsAutomaton</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Computes the right Cayley graph of a finite monoid or semigroup <var class="Arg">S</var>. It uses the <strong class="pkg">GAP</strong> buit-in function <code class="code">CayleyGraphSemigroup</code> to compute the Cayley Graph and returns it as an automaton without initial nor final states. (In this automaton state <code class="code">i</code> represents the element <code class="code">Elements(S)[i]</code>.) The <strong class="pkg">Automata</strong> package is used to this effect.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightCayleyGraphMonoidAsAutomaton</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function is a synonym of <code class="func">RightCayleyGraphAsAutomaton</code> (<a href="chap2_mj.html#X822983CD7F01B5EA"><span class="RefLink">2.4-1</span></a>).</p>
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