GAP (SgpViz) - Chapter 2: Basics

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2 Basics
2.1 Examples
2.2 Some attributes
2.2-1 HasCommutingIdempotents
2.2-2 IsInverseSemigroup
2.3 Some basic functions
2.3-1 PartialTransformation
2.3-2 ReduceNumberOfGenerators
2.3-3 SemigroupFactorization
2.3-4 GrahamBlocks
2.4 Cayley graphs
2.4-1 RightCayleyGraphAsAutomaton
2.4-2 RightCayleyGraphMonoidAsAutomaton

### 2 Basics

We give some examples of semigroups to be used later. We also describe some basic functions that are not directly available from GAP, but are useful for the purposes of this package.

#### 2.1 Examples

These are some examples of semigroups that will be used through this manual

    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 ]>

    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>

#### 2.2 Some attributes

These functions are semigroup attributes that get stored once computed.

##### 2.2-1 HasCommutingIdempotents

`‣ HasCommutingIdempotents`( M ) | ( attribute )

Tests whether the idempotents of the semigroup M commute.

##### 2.2-2 IsInverseSemigroup

`‣ IsInverseSemigroup`( S ) | ( attribute )

Tests whether a finite semigroup S is inverse. It is well-known that it suffices to test whether the idempotents of S commute and S is regular. The function `IsRegularSemigroup` is part of GAP.

#### 2.3 Some basic functions

##### 2.3-1 PartialTransformation

`‣ PartialTransformation`( L ) | ( function )

A partial transformation is a partial function of a set of integers of the form \(\{1,\dots, n\}\). It is given by means of the list of images L. 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.

    gap> PartialTransformation([2,0,4,0]);
    Transformation( [ 2, 5, 4, 5, 5 ] )

##### 2.3-2 ReduceNumberOfGenerators

`‣ ReduceNumberOfGenerators`( L ) | ( function )

Given a subset L of the generators of a semigroup, returns a list of generators of the same semigroup but possibly with less elements than L.

##### 2.3-3 SemigroupFactorization

`‣ SemigroupFactorization`( S, L ) | ( function )

L is an element (or list of elements) of the semigroup S. Returns a minimal factorization on the generators of S of the element(s) of L. Works only for transformation semigroups.

    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 ] ) ] ]

##### 2.3-4 GrahamBlocks

`‣ GrahamBlocks`( mat ) | ( function )

mat is a matrix as displayed by `DisplayEggBoxOfDClass(D);` of a regular D-class `D`. This function outputs a list `[gmat, phi]` where `gmat` is mat 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.

    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 ] ] ]

#### 2.4 Cayley graphs

##### 2.4-1 RightCayleyGraphAsAutomaton

`‣ RightCayleyGraphAsAutomaton`( S ) | ( function )

Computes the right Cayley graph of a finite monoid or semigroup S. It uses the GAP buit-in function `CayleyGraphSemigroup` to compute the Cayley Graph and returns it as an automaton without initial nor final states. (In this automaton state `i` represents the element `Elements(S)[i]`.) The Automata package is used to this effect.

    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: [ ]

##### 2.4-2 RightCayleyGraphMonoidAsAutomaton

`‣ RightCayleyGraphMonoidAsAutomaton`( S ) | ( function )

This function is a synonym of `RightCayleyGraphAsAutomaton` (2.4-1).

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