GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="ISO-8859-1"?>
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  <!-- $Id: schutzenberger.xml,v 0.998  Exp $ -->
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  <Chapter>
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    <Heading>User friendly ways to give semigroups and automata</Heading>
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This chapter describes two Tcl/Tk graphical interfaces that can be used to
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define and edit semigroups and automata. 
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<Section>
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      <Heading> 
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Finite automata 
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      </Heading>
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      <ManSection>
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        <Func Arg="[A]" Name="XAutomaton" />
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        <Description>
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The function <C> Xautomaton </C> without arguments opens a new window where
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an automaton may be specified. A finite automaton (which may then be edited)
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may be given as argument.  
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        </Description>
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      </ManSection>
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      <Example><![CDATA[
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gap> XAutomaton();
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      ]]></Example>
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It opens a window like the following:
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    <P />
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xaut.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xaut.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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<C> Var </C> is the &GAP; name of the automaton, <C> States </C> is the
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number of states, <C> Alphabet </C> represents the alphabet and may be given
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through a positive integer (in this case the alphabet is understood to be <C>
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a,b,c,... </C>) or through a string whose symbols, in order, being the letters
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of the alphabet. The numbers corresponding to the initial 
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and accepting states are placed in the respective boxes. The automaton may be
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specified to be deterministic, non deterministic or with epsilon
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transitions. After pressing the <B> transition matrix </B> button the window
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gets larger and the
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transition matrix of the automaton may be given.
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The <E>i</E>th row of the matrix describes the action of the
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<E>i</E>th letter on the states.
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xautoma.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xautoma.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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A non deterministic automaton may be given as follows:
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xndAUT.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xndAUT.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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<P/>
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By pressing the button <B> Ok </B> the &GAP; shell aquires the aspect shown
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in the following picture and the automaton <C> ndAUT </C> may be used to do
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computations. Some computations such as getting a rational expression
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representing the language of the automaton, the (complete) minimal automaton
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representing the same language or the transition semigroup of the automaton,
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may be done directly after pressing the <B> Functions</B> button.
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xndAUTok.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xndAUTok.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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<P/>
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By pressing the button <B> View </B> an image representing the automaton is
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displayed in a new window.
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/ndAutImage.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/ndAutImage.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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An automaton with epsilon transitions may be given as follows shown in the
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following picture. The last letter of the alphabet is always considered to be
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the <M> \epsilon</M>. In the images it is represented by @.
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/EpsTrAut.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/EpsTrAut.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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<P/>
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A new window with an image representing the automaton may be obtained by pressing the button <B> View </B>.
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/EpsTrAutImage.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/EpsTrAutImage.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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<P/>
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In the next example it is given an argument to the function <C>XAutomaton</C>.
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      <Example><![CDATA[
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gap> A := RandomAutomaton("det",2,2);
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< deterministic automaton on 2 letters with 2 states >
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gap> XAutomaton(A);
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      ]]></Example>
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It opens a window like the following:
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    <P />
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xautgiven.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xautgiven.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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     </Section>
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    <Section>
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    <Heading> Finite semigroups </Heading>
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The most common ways to give a semigroup to are through generators and
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relations, a set of (partial) transformations as generating set and as
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syntactic semigroups of automata or rational languages. 
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      <ManSection>
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        <Func Arg="[S]" Name="XSemigroup" />
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        <Description>
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The function <C> XSemigroup </C> without arguments opens a new window where
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a semigroup (or monoid) may be specified. A finite semigroup (which may then be edited) may be given as argument.  
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        </Description>
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      </ManSection>
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      <Example><![CDATA[
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gap> XSemigroup();
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      ]]></Example>
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It opens a window like the following:
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xsgp.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xsgp.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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where one may choose how to give the semigroup. 
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    <Subsection>
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      <Heading> 
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Semigroups given through generators and relations 
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      </Heading>
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In the window opened by <C>XSemigroup</C>, by pressing the button
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<B>Proceed</B> the window should enlarge and have the following aspect. 
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(If the window does not enlarge automatically, use the mouse to do it.) 
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    <P />
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xb211.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xb211.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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<C> GAP variable </C> is the &GAP; name of the semigroup. One has then to specify the
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number of generators, the number of relations (which does not to be exact)
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and whether one wants to produce a monoid or a semigroup.
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Pressing the <B>Proceed</B> button one gets:
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xb212.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xb212.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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     <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/xb21.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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When giving the relations, the usual abbreviations "0" and "1" may be used.
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<P/>
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Pressing the <B>Done</B> button would output the following to the shell where <Package>GAP</Package>
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is running:
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    </Alt>
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/gapb21.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/gapb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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    <Alt Only="LaTeX">
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<P/>
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The menu button <B>Functions</B> has the following commands:
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/funcsind.png}
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        \includegraphics[width=0.4\textwidth]{images/funcsmenu.png}
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        \end{center}
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The interface allows to add and remove  &GAP; functions to the menu. When
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adding a function, the name of the function should be provided.
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(In its current version, it works only with functions that have
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as only argument a semigroup.)
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<P/>
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By pressing the menu button <B>Functions</B> and selecting "Draw Schutzenberger Graphs" would
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pop up the following window:
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/ShuzenbergerGraphb21.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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<P/>
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The menu button <B>Functions</B> has the following commands:
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&#60;br&#62;&#60;center&#62;
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&#60;img src=&#34;images/funcsind.png&#34;&#62;
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&#60;img src=&#34;images/funcsmenu.png&#34;&#62;
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&#60;/center&#62;&#60;br&#62;
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The interface allows to add and remove  &GAP; functions to the menu. When
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adding a function, the name of the function should be provided.
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(In its current version, it works only with functions that have
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as only argument a semigroup.)
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<P/>
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By pressing the menu button <B>Functions</B> and selecting "Draw Schutzenberger Graphs" would
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pop up the following window:
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/ShuzenbergerGraphb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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    <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/CayleyGraphb21.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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By pressing the menu button <B>Functions</B> and selecting "Draw Cayley Graph" would
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pop up the following window:
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/CayleyGraphb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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     <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/DClassesb21.png}
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        \end{center}
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    </Alt>
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    <Alt Only="HTML">
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By pressing the menu button <B>Functions</B> and selecting "Draw D-Classes" would
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pop up the following window:
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/DClassesb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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     </Subsection>
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     <Subsection>
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      <Heading> 
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Semigroups given by partial transformations
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      </Heading>
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<Code>XSemigroup(poi3);</Code> would pop up the following window, where everything should be clear:
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     <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/Xsemigroup1.png}
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        \end{center}
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    </Alt>
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<Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/Xsemigroup1.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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    </Subsection>
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    <Subsection>
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      <Heading> 
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Syntatic semigroups
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      </Heading>
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<Code>XSemigroup();</Code> would pop up the following window, where we would select "Syntatic semigroup",
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press the <B>Proceed</B> button and then choose either to give a "Rational expression" or an "Automaton"
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by pressing one of those buttons:
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     <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/Xsemigroup2.png}
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        \end{center}
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    </Alt>
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<Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/Xsemigroup2.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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If "Rational expression" is chosen, a new window pops up where the expression can be specified:
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     <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/regexp.png}
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        \end{center}
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    </Alt>
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<Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/regexp.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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After pressing the <B>Ok</B> button, notice that the menu button <B>Functions</B>
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appears on the main window (lower right corner) meaning that <Package>GAP</Package>
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already recognizes the given semigroup:
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     <Alt Only="LaTeX">
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        \begin{center}
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        \includegraphics[width=0.4\textwidth]{images/regexp2.png}
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        \end{center}
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    </Alt>
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<Alt Only="HTML">
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      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/regexp2.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
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    </Alt>
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    </Subsection>
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    </Section>
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  </Chapter>
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