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<p><a id="X868F7BAB7AC2EEBC" name="X868F7BAB7AC2EEBC"></a></p>
<div class="ChapSects"><a href="chap2.html#X868F7BAB7AC2EEBC">2 <span class="Heading">Basics</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7A489A5D79DA9E5C">2.1 <span class="Heading">Examples </span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X85134313846D1A8A">2.2 <span class="Heading">Some attributes</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F7FAED380682973">2.2-1 HasCommutingIdempotents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83F1529479D56665">2.2-2 IsInverseSemigroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X78CA2A0D869C51DC">2.3 <span class="Heading">Some basic functions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A65787A83C0F8EF">2.3-1 PartialTransformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7EEED52C7D38E1CA">2.3-2 ReduceNumberOfGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7BBEBEE885D05208">2.3-3 SemigroupFactorization</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7FB7633483A45209">2.3-4 GrahamBlocks</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X789D5E5A8558AA07">2.4 <span class="Heading">Cayley graphs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X822983CD7F01B5EA">2.4-1 RightCayleyGraphAsAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X82F7D3E485D615D8">2.4-2 RightCayleyGraphMonoidAsAutomaton</a></span>
</div>
</div>
<h3>2 <span class="Heading">Basics</span></h3>
<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>
<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<h4>2.1 <span class="Heading">Examples </span></h4>
<p>These are some examples of semigroups that will be used through this manual</p>
<table class="example">
<tr><td><pre>
gap> f := FreeMonoid("a","b");
<free monoid on the generators [ a, b ]>
gap> a := GeneratorsOfMonoid( f )[ 1 ];;
gap> b := GeneratorsOfMonoid( f )[ 2 ];;
gap> r:=[[a^3,a^2],
> [a^2*b,a^2],
> [b*a^2,a^2],
> [b^2,a^2],
> [a*b*a,a],
> [b*a*b,b] ];
[ [ a^3, a^2 ], [ a^2*b, a^2 ], [ b*a^2, a^2 ], [ b^2, a^2 ], [ a*b*a, a ],
[ b*a*b, b ] ]
gap> b21:= f/r;
<fp semigroup on the generators [<identity ... >, a, b ]> </pre></td></tr></table>
<table class="example">
<tr><td><pre>
gap> f := FreeSemigroup("a","b");
<free semigroup on the generators [ a, b ]>
gap> a := GeneratorsOfSemigroup( f )[ 1 ];;
gap> b := GeneratorsOfSemigroup( f )[ 2 ];;
gap> r:=[[a^3,a^2],
> [a^2*b,a^2],
> [b*a^2,a^2],
> [b^2,a^2],
> [a*b*a,a],
> [b*a*b,b] ];
[ [ a^3, a^2 ], [ a^2*b, a^2 ], [ b*a^2, a^2 ], [ b^2, a^2 ], [ a*b*a, a ],
[ b*a*b, b ] ]
gap> b2:= f/r;
<fp semigroup on the generators [ a, b ]> </pre></td></tr></table>
<table class="example">
<tr><td><pre>
gap> g0:=Transformation([4,1,2,4]);;
gap> g1:=Transformation([1,3,4,4]);;
gap> g2:=Transformation([2,4,3,4]);;
gap> poi3:= Monoid(g0,g1,g2);
<monoid with 3 generators>
</pre></td></tr></table>
<p><a id="X85134313846D1A8A" name="X85134313846D1A8A"></a></p>
<h4>2.2 <span class="Heading">Some attributes</span></h4>
<p>These functions are semigroup attributes that get stored once computed.</p>
<p><a id="X7F7FAED380682973" name="X7F7FAED380682973"></a></p>
<h5>2.2-1 HasCommutingIdempotents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> HasCommutingIdempotents</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Tests whether the idempotents of the semigroup <var class="Arg">M </var>commute.</p>
<p><a id="X83F1529479D56665" name="X83F1529479D56665"></a></p>
<h5>2.2-2 IsInverseSemigroup</h5>
<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>
<p><a id="X78CA2A0D869C51DC" name="X78CA2A0D869C51DC"></a></p>
<h4>2.3 <span class="Heading">Some basic functions</span></h4>
<p><a id="X7A65787A83C0F8EF" name="X7A65787A83C0F8EF"></a></p>
<h5>2.3-1 PartialTransformation</h5>
<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 {1, ..., n}. 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>
<table class="example">
<tr><td><pre>
gap> PartialTransformation([2,0,4,0]);
Transformation( [ 2, 5, 4, 5, 5 ] )
</pre></td></tr></table>
<p><a id="X7EEED52C7D38E1CA" name="X7EEED52C7D38E1CA"></a></p>
<h5>2.3-2 ReduceNumberOfGenerators</h5>
<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>
<p><a id="X7BBEBEE885D05208" name="X7BBEBEE885D05208"></a></p>
<h5>2.3-3 SemigroupFactorization</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SemigroupFactorization</code>( <var class="Arg">SL</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>
<table class="example">
<tr><td><pre>
gap> el1 := Transformation( [ 2, 3, 4, 4 ] );;
gap> el2 := Transformation( [ 2, 4, 3, 4 ] );;
gap> f1 := SemigroupFactorization(poi3,el1);
[ [ Transformation( [ 1, 3, 4, 4 ] ), Transformation( [ 2, 4, 3, 4 ] ) ] ]
gap> f1[1][1] * f1[1][2] = el1;
true
gap> SemigroupFactorization(poi3,[el1,el2]);
[ [ Transformation( [ 1, 3, 4, 4 ] ), Transformation( [ 2, 4, 3, 4 ] ) ],
[ Transformation( [ 2, 4, 3, 4 ] ) ] ]
</pre></td></tr></table>
<p><a id="X7FB7633483A45209" name="X7FB7633483A45209"></a></p>
<h5>2.3-4 GrahamBlocks</h5>
<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 <code class="code">phi</code> maps H-classes of <code class="code">gmat</code> to the corresponding ones of <var class="Arg">mat</var>, i.e., <code class="code">phi[i][j] = [i',j']</code> where <code class="code">mat[i'][j'] = gmat[i][j]</code>. If the argument to this function is not a matrix corresponding to a regular D-class, the function may abort in error.</p>
<table class="example">
<tr><td><pre>
gap> p1 := PartialTransformation([6,2,0,0,2,6,0,0,10,10,0,0]);;
gap> p2 := PartialTransformation([0,0,1,5,0,0,5,9,0,0,9,1]);;
gap> p3 := PartialTransformation([0,0,3,3,0,0,7,7,0,0,11,11]);;
gap> p4 := PartialTransformation([4,4,0,0,8,8,0,0,12,12,0,0]);;
gap> css3:=Semigroup(p1,p2,p3,p4);
<semigroup with 4 generators>
gap> el := Elements(css3)[8];;
gap> D := GreensDClassOfElement(css3, el);;
gap> IsRegularDClass(D);
true
gap> DisplayEggBoxOfDClass(D);
[ [ 1, 0, 1, 0 ],
[ 0, 1, 0, 1 ],
[ 0, 1, 0, 1 ],
[ 1, 0, 1, 0 ] ]
gap> mat := [ [ 1, 0, 1, 0 ],
> [ 0, 1, 0, 1 ],
> [ 0, 1, 0, 1 ],
> [ 1, 0, 1, 0 ] ];;
gap> res := GrahamBlocks(mat);;
gap> PrintArray(res[1]);
[ [ 1, 1, 0, 0 ],
[ 1, 1, 0, 0 ],
[ 0, 0, 1, 1 ],
[ 0, 0, 1, 1 ] ]
gap> PrintArray(res[2]);
[ [ [ 1, 1 ], [ 1, 3 ], [ 1, 2 ], [ 1, 4 ] ],
[ [ 4, 1 ], [ 4, 3 ], [ 4, 2 ], [ 4, 4 ] ],
[ [ 2, 1 ], [ 2, 3 ], [ 2, 2 ], [ 2, 4 ] ],
[ [ 3, 1 ], [ 3, 3 ], [ 3, 2 ], [ 3, 4 ] ] ]
</pre></td></tr></table>
<p><a id="X789D5E5A8558AA07" name="X789D5E5A8558AA07"></a></p>
<h4>2.4 <span class="Heading">Cayley graphs</span></h4>
<p><a id="X822983CD7F01B5EA" name="X822983CD7F01B5EA"></a></p>
<h5>2.4-1 RightCayleyGraphAsAutomaton</h5>
<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>
<table class="example">
<tr><td><pre>
gap> rcg := RightCayleyGraphAsAutomaton(b21);
< deterministic automaton on 2 letters with 6 states >
gap> Display(rcg);
| 1 2 3 4 5 6
-----------------------
a | 2 4 6 4 2 4
b | 3 5 4 4 4 3
Initial state: [ ]
Accepting state: [ ]
</pre></td></tr></table>
<p><a id="X82F7D3E485D615D8" name="X82F7D3E485D615D8"></a></p>
<h5>2.4-2 RightCayleyGraphMonoidAsAutomaton</h5>
<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.html#X822983CD7F01B5EA"><b>2.4-1</b></a>).</p>
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