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  <Chapter>
    <Heading>Basics</Heading>
We give some examples of semigroups to be used later. We also describe some basic functions
that are not directly available from <Package>GAP</Package>, but are useful
for the purposes of this package.
    <Section>
      <Heading>Examples </Heading>These are some examples of semigroups that will be   used through this manual
      <Example><![CDATA[
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 monoid on the generators [ a, b ]>
      ]]></Example>

     <Example><![CDATA[
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>
     ]]></Example>

    </Section>

    <Section>
      <Heading>Some attributes</Heading>
These functions are semigroup attributes that get stored once computed.
      <ManSection>
        <Attr Arg="M" Name="HasCommutingIdempotents" />
        <Description>Tests whether the idempotents of the semigroup
          <Arg>M </Arg>commute. </Description>
      </ManSection>
      <ManSection>
        <Attr Arg="S" Name="IsInverseSemigroup" />
        <Description>Tests whether a finite semigroup
          <Arg>S </Arg>is inverse.       It is well-known that it suffices to test whether the idempotents of
          <Arg>S </Arg>commute and
          <Arg>S </Arg>is regular. The function
          <Code>IsRegularSemigroup </Code>is part of
          <Package>GAP</Package>. </Description>
      </ManSection>
    </Section>

    <Section>
      <Heading>Some basic functions</Heading>
      <ManSection>
        <Func Arg="L" Name="PartialTransformation" />
        <Description>A partial transformation is a partial function of a set of integers of
the form  <M>\{1,\dots, n\}</M>. It is given by means of the list of images <Arg>L</Arg>. 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.</Description>
      </ManSection>

      <Example><![CDATA[
gap> PartialTransformation([2,0,4,0]);
Transformation( [ 2, 5, 4, 5, 5 ] )
      ]]></Example>

      <ManSection>
        <Func Arg="L" Name="ReduceNumberOfGenerators" />
        <Description>Given a subset <Arg>L</Arg> of the generators
of a semigroup, returns a list of generators of the same semigroup but
possibly with less elements than <Arg>L</Arg>.</Description>
      </ManSection>

      <ManSection>
        <Func Arg="S,L" Name="SemigroupFactorization" />
        <Description><Arg>L</Arg> is an element (or list of elements) of the semigroup <Arg>S</Arg>.
Returns a minimal factorization on the generators of <Arg>S</Arg> of the
element(s) of <Arg>L</Arg>. Works only for transformation semigroups.</Description>
      </ManSection>
        <Example><![CDATA[
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 ] ) ] ]
]]></Example>

    <ManSection>
        <Func Arg="mat" Name="GrahamBlocks" />
        <Description><Arg>mat</Arg> is a matrix as displayed by
<C>DisplayEggBoxOfDClass(D);</C> of a regular D-class <C>D</C>.
This function outputs a list <C>[gmat, phi]</C> where <C>gmat</C> is
<Arg>mat</Arg> in Graham's blocks form and phi maps H-classes
of <C>gmat</C> to the corresponding ones of <Arg>mat</Arg>, i.e.,
<C>phi[i][j] = [i',j']</C> where <C>mat[i'][j'] = gmat[i][j]</C>.
If the argument to this function is not a matrix corresponding to
a regular D-class, the function may abort in error.
      </Description>
      </ManSection>
      <Example><![CDATA[
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);
<transformation semigroup of degree 13 with 4 generators>
gap> el := Elements(css3)[8];;
gap> D := GreensDClassOfElement(css3, el);;
gap> IsRegularDClass(D);
true
gap> DisplayEggBoxOfDClass(D);
[ [  1,  1,  0,  0 ],
  [  1,  1,  0,  0 ],
  [  0,  0,  1,  1 ],
  [  0,  0,  1,  1 ] ]
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 ] ] ]
]]></Example>
      
    </Section>

    <Section>
      <Heading>Cayley graphs</Heading>
      <ManSection>
        <Func Arg="S" Name="RightCayleyGraphAsAutomaton"></Func>
        <Description>Computes the right Cayley graph of a finite monoid or semigroup <Arg>S</Arg>. It uses the
<Package>GAP</Package>
buit-in function <Code>CayleyGraphSemigroup</Code> to compute the Cayley Graph
and returns it as an automaton without initial nor final states.
(In this automaton state <C>i</C> represents the element <C>Elements(S)[i]</C>.)
The <Package>Automata</Package> package is used to this effect.
          </Description>
      </ManSection>

      <Example><![CDATA[
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: [ ]
      ]]></Example>

      <ManSection>
        <Func Arg="S" Name="RightCayleyGraphMonoidAsAutomaton" />
        <Description>This function is a synonym of
<Ref Func="RightCayleyGraphAsAutomaton"/>.
          </Description>
      </ManSection>



    </Section>

  </Chapter>

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