GAP (FR) - Chapter 11: Miscellanea

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11 Miscellanea
11.1 Generic operations
11.1-1 TensorSum
11.1-2 TensorProduct
11.1-3 DirectSum
11.2 Periodic lists
11.2-1 PeriodicListsFamily
11.2-2 PeriodicList
11.2-3 CompressPeriodicList
11.2-4 IsConfinal
11.2-5 ConfinalityClass
11.2-6 LargestCommonPrefix
11.3 Word growth
11.3-1 WordGrowth
11.4 Finding short relations
11.4-1 ShortGroupRelations
11.4-2 ShortGroupWordInSet
11.5 Braid groups
11.5-1 SurfaceBraidFpGroup
11.5-2 CharneyBraidFpGroup
11.5-3 ArtinRepresentation
11.6 Dirichlet series
11.6-1 DirichletSeries
11.6-2 DegreeDirichletSeries
11.6-3 SpreadDirichletSeries
11.6-4 ShiftDirichletSeries
11.6-5 ShrunkDirichletSeries
11.6-6 ZetaSeriesOfGroup
11.6-7 ValueOfDirichletSeries
11.7 Projective representations
11.7-1 IsProjectiveRepresentation
11.7-2 ProjectiveRepresentationByFunction
11.7-3 LinearRepresentationByImages
11.7-4 DegreeOfProjectiveRepresentation
11.7-5 ProjectiveExtension
11.7-6 ProjectiveQuotient
11.8 Miscellanea
11.8-1 ForwardOrbit
11.8-2 StringByInt
11.8-3 PositionInTower
11.8-4 RenameSubobjects
11.8-5 CoefficientsInAbelianExtension
11.8-6 MagmaEndomorphismByImagesNC
11.8-7 MagmaHomomorphismByImagesNC
11.8-8 Draw
11.8-9 IsFIFO
11.8-10 ProductIdeal
11.8-11 DimensionSeries
11.8-12 AlgebraHomomorphismByFunction
11.8-13 IsFpLieAlgebra
11.8-14 JenningsLieAlgebra
11.8-15 SolutionMatModN
11.8-16 SolutionMatMod1
11.8-17 CyclotomicByArgument
11.8-18 ArgumentOfCyclotomic
11.9 User settings
11.9-1 InfoFR
11.9-2 SEARCH@

### 11 Miscellanea

#### 11.1 Generic operations

##### 11.1-1 TensorSum

`‣ TensorSum`( objects, ... ) | ( function )

This function is similar in syntax to `DirectProduct` (Reference: DirectProduct), and delegates to `TensorSumOp`; its meaning depends on context, see e.g. `TensorSumOp` (3.5-4).

##### 11.1-2 TensorProduct

`‣ TensorProduct`( objects, ... ) | ( function )

This function is similar in syntax to `DirectProduct` (Reference: DirectProduct), and delegates to `TensorProductOp`; its meaning depends on context, see e.g. `TensorProductOp` (3.5-5).

##### 11.1-3 DirectSum

`‣ DirectSum`( objects, ... ) | ( function )

This function is similar in syntax to `DirectProduct` (Reference: DirectProduct), and delegates to `DirectSumOp`; its meaning depends on context, see e.g. `DirectSumOp` (3.5-6).

#### 11.2 Periodic lists

##### 11.2-1 PeriodicListsFamily

`‣ PeriodicListsFamily` | ( family )

`‣ IsPeriodicList` | ( filter )

The family, respectively filter, of `PeriodicList` (11.2-2)s.

##### 11.2-2 PeriodicList

`‣ PeriodicList`( preperiod[, period] ) | ( operation )

`‣ PeriodicList`( list, i ) | ( operation )

`‣ PeriodicList`( list, f ) | ( operation )

`‣ CompressedPeriodicList`( preperiod[, period] ) | ( operation )

`‣ CompressedPeriodicList`( list, i ) | ( operation )

`‣ PrePeriod`( list ) | ( operation )

`‣ Period`( list ) | ( operation )

These functions manipulate periodic lists, i.e. lists of infinite length such that elements follow a periodic order after some point.

The first command creates a periodic list, specified by its preperiod and period, which must both be lists. If the period is absent, this is actually a finite list.

The second command creates a periodic list by decreeing that the entries after the end of the list start again at position i.

The third command creates a list by applying function f to all elements of l.

The fourth and fifth command compress the newly created periodic list, see `CompressPeriodicList` (11.2-3).

The sixth and seventh commands return respectively the preperiod and period of a periodic list.

Most of the methods applied for lists have an obvious equivalent for periodic lists: `List` (Reference: list element operation), `Filtered` (Reference: Filtered), `First` (Reference: First), `ForAll` (Reference: ForAll), `ForAny` (Reference: ForAny), `Number` (Reference: Number).

    gap> l := PeriodicList([1],[2,3,4]);
    [ 1, / 2, 3, 4 ]
    gap> l[5];
    2
    gap> Add(l,100,3); l;
    [ 1, 2, 100, / 3, 4, 2 ]
    gap> Remove(l,5);
    4
    gap> l;
    [ 1, 2, 100, 3, / 2, 3, 4 ]
    gap> PrePeriod(l);
    [ 1, 2, 100, 3 ]
    gap> Period(l);
    [ 2, 3, 4 ]

##### 11.2-3 CompressPeriodicList

`‣ CompressPeriodicList`( l ) | ( operation )

This function compresses a periodic list, in replacing the period by a minimal period, and shortening the preperiod. No value is returned, but the list l is modified. It remains equal (under `=`) to the original list.

    gap> l := PeriodicList([1],[2,3,4,2,3,4]);
    [ 1, / 2, 3, 4, 2, 3, 4 ]
    gap> Add(l,4,5); l;
    [ 1, 2, 3, 4, 4, / 2, 3, 4, 2, 3, 4 ]
    gap> CompressPeriodicList(l);
    gap> l;
    [ 1, 2, 3, 4, / 4, 2, 3 ]

##### 11.2-4 IsConfinal

`‣ IsConfinal`( l, m ) | ( operation )

Returns: `true` if l and m are eventually equal.

This function tests whether two lists are confinal, i.e. whether, after removal of the same suitable number of elements from both lists, they become equal.

    gap> l := PeriodicList([1],[2,3,2,3]);
    [ 1, / 2, 3, 2, 3 ]
    gap> m := PeriodicList([0,1],[3,2]);
    [ 0, 1, / 3, 2 ]
    gap> IsConfinal(l,m);
    true

##### 11.2-5 ConfinalityClass

`‣ ConfinalityClass`( l ) | ( operation )

Returns: The strictly periodic list with same tail as l.

There exists a unique periodic list, with no preperiod, which is confinal (see `IsConfinal` (11.2-4)) to l. This strictly periodic list is returned by this command.

    gap> l := PeriodicList([1],[2,3,2,3]);
    [ 1, / 2, 3, 2, 3 ]
    gap> ConfinalityClass(l);
    [/ 3, 2 ]

##### 11.2-6 LargestCommonPrefix

`‣ LargestCommonPrefix`( c ) | ( operation )

Returns: The longest list that is a prefix of all elements of c.

This command computes the longest (finite or periodic) list which is a prefix of all elements of c. The argument c is a collection of finite and periodic lists.

    gap> LargestCommonPrefix([PeriodicList([1],[2,3,2,3]),[1,2,3,4]]);
    [ 1, 2, 3 ]

#### 11.3 Word growth

##### 11.3-1 WordGrowth

`‣ WordGrowth`( g, rec(options...) ) | ( function )

`‣ WordGrowth`( g: options... ) | ( function )

`‣ OrbitGrowth`( g, point[, limit] ) | ( function )

`‣ Ball`( g, radius ) | ( function )

`‣ Sphere`( g, radius ) | ( function )

Returns: The word growth of the semigroup g.

This function computes the first terms of growth series associated with the semigroup g. The argument g can actually be a group/monoid/semigroup, or a list representing that semigroup's generating set.

The behaviour of `WordGrowth` is controlled via options passed in the second argument, which is a record. They can be combined when reasonable, and are:

`limit:=n`

  to specify a limit radius;

`sphere:=radius`

  to return the sphere of the specified radius, unless a radius was specified in `limit`, in which case the value is ignored;

`spheres:=maxradius`

  to return the list of spheres of radius between 0 and the specified limit;

`spheresizes:=maxradius`

  to return the list sizes of spheres of radius between 0 and the specified limit;

`ball:=radius`

  to return the ball of the specified radius;

`balls:=maxradius`

  to return the list of balls of radius between 0 and the specified limit;

`ballsizes:=maxradius`

  to return the list sizes of balls of radius between 0 and the specified limit;

`indet:=z`

  to return the `spheresizes`, as a polynomial in `z` (or the first indeterminate if `z` is not a polynomial;

`draw:=filename`

  to create a rendering of the Cayley graph of g. Edges are given colours according to the cyclic ordering "red", "blue", "green", "gray", "yellow", "cyan", "orange", "purple". If `filename` is a string, the graph is appended, in `dot` format, to that file. Otherwise, the output is converted to Postscript using the program `neato` from the graphviz package, and displayed in a separate X window using the program display or rsvg-view. This works on UNIX systems.

  It is assumed, but not checked, that graphviz and display/rsvg-view are properly installed on the system. The option `usesvg` requests the use of rsvg-view; by default, display is used.

`point:=p`

  to compute the growth of the orbit of `p` under g, rather than the growth of g.

`act:=function(point,g) ... end`

  to compute the action with a specific function, rather than the default `OnPoints`.

`track:=true`

  to keep track of a word in the generators that gives the element. This affects the "ball", "balls", "sphere" and "spheres" commands, where the result returned is a 3-element list: the first entry is the original results; the second entry is a homomorphism from a free group/monoid/semigroup; and the third entry contains the words corresponding to the first entry via the homomorphism.

If the first argument is an integer `n` and not a record, the command is interpreted as `WordGrowth(...,rec(spheresizes:=n))`.

`WordGrowth(...,rec(draw:=true))` may be abbreviated as `Draw(...)`; `WordGrowth(...,rec(ball:=n))` may be abbreviated as `Ball(...,n)`; `WordGrowth(...,rec(sphere:=n))` may be abbreviated as `Sphere(...,n)`;

    gap> WordGrowth(GrigorchukGroup,4);
    [ 1, 4, 6, 12, 17 ]
    gap> WordGrowth(GrigorchukGroup,rec(limit:=4,indet:=true));
    17*x_1^4+12*x_1^3+6*x_1^2+4*x_1+1
    gap> WordGrowth(GrigorchukGroup,rec(limit:=1,spheres:=true));
    [ [ <Mealy element on alphabet [ 1, 2 ] with 1 state, initial state 1> ],
      [ d, b, c, a ] ]
    gap> WordGrowth(GrigorchukGroup,rec(point:=[2,2,2]));
    [ 1, 1, 1, 1, 1, 1, 1, 1 ]
    gap> OrbitGrowth(GrigorchukGroup,[1,1,1]);
    [ 1, 2, 2, 1, 1, 1 ]
    gap> WordGrowth(GrigorchukGroup,rec(spheres:=4,point:=PeriodicList([],[2])));
    [ [ [/ 2 ] ], [ [ 1, / 2 ] ], [ [ 1, 1, / 2 ] ], [ [ 2, 1, / 2 ] ],
      [ [ 2, 1, 1, / 2 ] ] ]
    gap> WordGrowth([(1,2),(2,3)],rec(spheres:=infinity,track:=true));
    [ [ [  ], [ (2,3), (1,2) ], [ (), (1,2,3), (1,3,2) ], [ (1,3) ] ],
      MappingByFunction( <free semigroup on the generators [ s1, s2 ]>, <group>, function( w ) ... end ),
      [ [  ], [ s2, s1 ], [ s2^2, s2*s1, s1*s2 ], [ s2*s1*s2 ] ] ]

Note that the orbit growth of `[/2]` is constant 1, while that of `[/1]` is constant 2. The following code would find the point with maximal orbit growth of a semigroup acting on the integers (for example, constructed with `PermGroup` (7.2-1)):

    MaximalOrbitGrowth := function(g)
        local maxpt, growth, max;
        maxpt := LargestMovedPoint(g);
        growth := List([1..maxpt],n->WordGrowth(g:point:=n));
        max := Maximum(growth);
        return [max,Filtered([1..maxpt],n->growth[n]=max)];
    end;

For example, the command `Draw(BasilicaGroup,rec(point:=PeriodicList([],[2,1]),limit:=3));` produces (in a new window) the following picture:

#### 11.4 Finding short relations

##### 11.4-1 ShortGroupRelations

`‣ ShortGroupRelations`( g, n ) | ( operation )

`‣ ShortMonoidRelations`( g, n ) | ( operation )

Returns: A list of relations between words over g, of length at most n.

This function assumes that g is a list of monoid elements. it searches for products of at most n elements over g that are equal.

In its first form, it returns a list of words in a free group `f` of rank the length of g, that are trivial in g. The first argument may be a group, in which case its symmetric generating set is considered.

In its second form, it returns a list of pairs `[l,r]`, where `l` and `r` are words in a free monoid `f` of rank the length of g, that are equal in g. The first argument may be a monoid, in which case its monoid generating set is considered.

This command does not construct all such pairs; rather, it returns a small set, in the hope that it may serve as a presentation for the monoid generated by g.

The first element of the list returned is actually not a relation: it is a homomorphism from `f` to [the group/monoid generated by] g.

    gap> ShortGroupRelations(GrigorchukGroup,10);
    [ [ x1, x2, x3, x4 ] -> [ a, b, c, d ],
      x1^2, x2^2, x3^2, x4^2, x2*x3*x4, x4*x1*x4*x1*x4*x1*x4*x1,
      x3*x1*x3*x1*x3*x1*x3*x1*x3*x1*x3*x1*x3*x1*x3*x1 ]
    gap> ShortGroupRelations(GuptaSidkiGroup,9);
    [ [ x1, x2 ] -> [ x, gamma ],
      x1^3, x2^3, x2*x1^-1*x2*x1^-1*x2*x1^-1*x2*x1^-1*x2*x1^-1*x2*x1^-1*
         x2*x1^-1*x2*x1^-1*x2*x1^-1,    x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*
    x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1 ]

##### 11.4-2 ShortGroupWordInSet

`‣ ShortGroupWordInSet`( g, s, n ) | ( operation )

`‣ ShortMonoidWordInSet`( g, s, n ) | ( operation )

`‣ ShortSemigroupWordInSet`( g, s, n ) | ( operation )

Returns: Words over g that express elements of s.

This command produces words in the free group/monoid/semigroup generated by g's generators that express elements of the set s. Elements of length at most AbsoluteValue(n) are searched; if n is non-negative then at most one element is returned. The value `n=infinity` is allowed.

The second argument may be either a list, a predicate (i.e. a function returning `true` or `false`) or an element.

The function returns a list of words in the free group/monoid/semigroup; the first entry of the list is a homomorphism from the free group/monoid/semigroup to g.

    gap> l := ShortMonoidWordInSet(Group((1,2),(2,3),(3,4)),
                [(1,2,3,4),(4,3,2,1)],-3);
    [ MappingByFunction( <free monoid on the generators [ m1, m2, m3 ]>, Group(
        [ (1,2), (2,3), (3,4) ]), function( w ) ... end ), m3*m2*m1, m1*m2*m3 ]
    gap> f := Remove(l,1);;
    gap> List(l,x->x^f);
    [ (1,2,3,4), (1,4,3,2) ]
    gap> ShortMonoidWordInSet(GrigorchukGroup,
           [Comm(GrigorchukGroup.1,GrigorchukGroup.2)],4);
    [ MappingByFunction( <free monoid on the generators [ m1, m2, m3, m4
         ]>, <self-similar monoid over [ 1 .. 2 ] with
        4 generators>, function( w ) ... end ), m1*m2*m1*m2 ]

#### 11.5 Braid groups

##### 11.5-1 SurfaceBraidFpGroup

`‣ SurfaceBraidFpGroup`( n, g, p ) | ( function )

`‣ PureSurfaceBraidFpGroup`( n, g, p ) | ( function )

Returns: The [pure] surface braid group on n strands.

This function creates a finitely presented group, isomorphic to the [pure] braid group on n strands of the surface of genus g, with p punctures. In particular, `SurfaceBraidFpGroup(n,0,1)` is the usual braid group (on the disc).

The presentation comes from [Bel04]. The first \(2g\) generators are the standard \(a_i,b_i\) surface generators; the next \(n-1\) are the standard \(s_i\) braid generators; and the last are the extra \(z\) generators.

The pure surface braid group is the kernel of the natural map from the surface braid group to the symmetric group on n points, defined by sending \(a_i,b_i,z\) to the identity and \(s_i\) to the transposition `(i,i+1)`.

##### 11.5-2 CharneyBraidFpGroup

`‣ CharneyBraidFpGroup`( n ) | ( function )

Returns: The braid group on n strands.

This function creates a finitely presented group, isomorphic to the braid group on n strands (on the disc). It is isomorphic to `SurfaceBraidFpGroup(n,0,1)`, but has a different presentation, due to Charney ([Cha95]), with one generator per non-trivial permutation of n points.

##### 11.5-3 ArtinRepresentation

`‣ ArtinRepresentation`( n ) | ( function )

Returns: The braid group's representation on `FreeGroup(n)`.

This function creates a Artin's representatin, a homomorphism from the braid group on n strands (on the disc) into the automorphism group of a free group of rank n.

#### 11.6 Dirichlet series

##### 11.6-1 DirichletSeries

`‣ DirichletSeries`( ) | ( operation )

`‣ DirichletSeries`( maxdeg ) | ( operation )

`‣ DirichletSeries`( indices, coeffs[, maxdeg] ) | ( operation )

`‣ DirichletSeries`( series, maxdeg ) | ( operation )

Creates a new Dirichlet series, namely, a formal power series of the form \(f(s)=\sum_{n\ge1} a(n) n^{-s}\). Such series have a maximal degree, which may be `infinity`, and may be added or multiplied as polynomials.

##### 11.6-2 DegreeDirichletSeries

`‣ DegreeDirichletSeries`( f ) | ( attribute )

Returns: The maximal degree of a non-zero coefficient of f.

##### 11.6-3 SpreadDirichletSeries

`‣ SpreadDirichletSeries`( f, n ) | ( attribute )

Returns: The series \(f(ns)\).

##### 11.6-4 ShiftDirichletSeries

`‣ ShiftDirichletSeries`( s, n ) | ( attribute )

Returns: The series \(n^{-s}f(s)\).

##### 11.6-5 ShrunkDirichletSeries

`‣ ShrunkDirichletSeries`( f ) | ( attribute )

Returns: The series f, with maximal precision set to its maximal degree.

##### 11.6-6 ZetaSeriesOfGroup

`‣ ZetaSeriesOfGroup`( G ) | ( attribute )

Returns: The series \sum_{\chi\in\widehat G}(\dim G)^{-s}.

##### 11.6-7 ValueOfDirichletSeries

`‣ ValueOfDirichletSeries`( f, s ) | ( attribute )

Returns: The evaluation of f at s. Synonym for `Value`.

#### 11.7 Projective representations

##### 11.7-1 IsProjectiveRepresentation

`‣ IsProjectiveRepresentation`( rep ) | ( property )

`‣ IsLinearRepresentation`( rep ) | ( property )

A projective representation is a mapping to matrices, that is multiplicative up to scalars. This property is set by the following functions that create projective representations.

The second property describes those projective representations that are in fact homomorphisms.

##### 11.7-2 ProjectiveRepresentationByFunction

`‣ ProjectiveRepresentationByFunction`( group, matrixgroup, function ) | ( operation )

Returns: A projective representation of group.

##### 11.7-3 LinearRepresentationByImages

`‣ LinearRepresentationByImages`( group, matrixgroup, src, dst ) | ( operation )

Returns: A linear representation of group.

##### 11.7-4 DegreeOfProjectiveRepresentation

`‣ DegreeOfProjectiveRepresentation`( rep ) | ( operation )

Returns: The dimension of the matrices in the image of rep.

##### 11.7-5 ProjectiveExtension

`‣ ProjectiveExtension`( rep, group ) | ( operation )

Returns: A projective representation of group whose restriction to Source(rep) (which is a subgroup of group) is rep.

##### 11.7-6 ProjectiveQuotient

`‣ ProjectiveQuotient`( rep, hom ) | ( operation )

Returns: A projective representation of Image(hom) that comes from rep.

#### 11.8 Miscellanea

##### 11.8-1 ForwardOrbit

`‣ ForwardOrbit`( g, x ) | ( operation )

Returns: The forward orbit of x under g.

This operation computes the smallest list containing x and closed under application of g.

    gap> ForwardOrbit((1,2,4,3),2);
    [ 2, 4, 3, 1 ]
    gap> ForwardOrbit(Transformation([4,4,5,2,1]),1);
    [ 1, 4, 2 ]

##### 11.8-2 StringByInt

`‣ StringByInt`( n[, b] ) | ( function )

Returns: A string representing n in base b.

This function converts a positive integer to string. It accepts an optional second argument, which is a base in which to print n. By default, b is 2.

##### 11.8-3 PositionInTower

`‣ PositionInTower`( t, x ) | ( function )

Returns: The largest index such that `t[i]` contains x.

This function assumes t is a descending tower of domains, such as that constructed by `LowerCentralSeries`. It returns the largest integer `i` such that `t[i]` contains x; in case the tower ends precisely with x, the value `infinity` is returned.

x can be an element or a subdomain of `t[1]`.

##### 11.8-4 RenameSubobjects

`‣ RenameSubobjects`( obj, refobj ) | ( function )

This function traverses obj if it is a list or a record, and, when it finds an element which has no name, but is equal (in the sense of `=`) to an element of refobj, assigns it the name of that element.

    gap> trivial := Group(());; SetName(trivial,"trivial");
    gap> a := List([1..10],i->Group(Random(SymmetricGroup(3))));
    [ Group([ (2,3) ]), Group([ (2,3) ]), Group([ (1,3) ]), Group([ (1,3) ]),
      Group([ (1,3,2) ]), Group([ (1,3,2) ]), Group([ (1,2) ]), Group(()),
      Group([ (2,3) ]), Group([ (1,3,2) ]) ]
    gap> RenameSubobjects(a,[trivial]); a;
    [ Group([ (2,3) ]), Group([ (2,3) ]), Group([ (1,3) ]), Group([ (1,3) ]),
      Group([ (1,3,2) ]), Group([ (1,3,2) ]), Group([ (1,2) ]), trivial,
      Group([ (2,3) ]), Group([ (1,3,2) ]) ]

##### 11.8-5 CoefficientsInAbelianExtension

`‣ CoefficientsInAbelianExtension`( x, b, G ) | ( function )

Returns: The coefficients in b of the element x, modulo G.

If b is a list of group elements \(b_1,\ldots,b_k\), and \(H=\langle G,b_1,\ldots,b_k\rangle\) contains G as a normal subgroup, and \(H/G\) is abelian and \(x\in H\), then this function computes exponents \(e_1,\ldots,e_k\) such that \(\prod b_i^{e_i}G=xG\).

##### 11.8-6 MagmaEndomorphismByImagesNC

`‣ MagmaEndomorphismByImagesNC`( f, im ) | ( function )

Returns: An endomorphism of f.

This function constructs an endomorphism of the group,monoid or semigroup f specified by sending generator number \(i\) to the \(i\)th entry in im. It is a shortcut for a call to `GroupHomomorphismByImagesNC` or `MagmaHomomorphismByFunctionNC(...,MappedWord(...))`.

##### 11.8-7 MagmaHomomorphismByImagesNC

`‣ MagmaHomomorphismByImagesNC`( f, g, im ) | ( function )

Returns: An homomorphism from f to g.

This function constructs a homomorphism of the group,monoid or semigroup f specified by sending generator number \(i\) to the \(i\)th entry in im. It is a shortcut for a call to `GroupHomomorphismByImagesNC` or `MagmaHomomorphismByFunctionNC(...,MappedWord(...))`.

##### 11.8-8 Draw

`‣ Draw`( p ) | ( function )

`‣ HeightOfPoset`( p ) | ( function )

Returns: The length of a maximal chain in the poset.

##### 11.8-9 IsFIFO

`‣ IsFIFO` | ( filter )

`‣ NewFIFO`( [l] ) | ( operation )

`‣ Add`( f, i ) | ( operation )

`‣ Append`( f, l ) | ( operation )

These functions create and extend FIFOs, i.e. first-in first-out data structures.

The first command creates a FIFO, with an optional list initializing it.

The second and third commands add an element, or append a list, to the FIFO.

Elements are removed via `NextIterator(f)`, and the FIFO is tested for emptyness via `IsDoneIterator(f)`. Thus, a typical use is the following code, which tests in breadth-first manner that all numbers in `[1..1000]` have a successor which is prime:

    gap> f := NewFIFO([1..10000]);
    <iterator>
    gap> for i in f do if not IsPrime(i) then Add(f,i+1); fi; od;

##### 11.8-10 ProductIdeal

`‣ ProductIdeal`( a, b ) | ( function )

`‣ ProductBOIIdeal`( a, b ) | ( function )

Returns: the product of the ideals a and b.

The first command computes the product of the left ideal a and the right ideal b. If they are not appropriately-sided ideals, the command first attempts to convert them.

The second command assumes that the ring of these ideals has a basis made of invertible elements. It is then much easier to compute the product.

##### 11.8-11 DimensionSeries

`‣ DimensionSeries`( a[, n] ) | ( function )

Returns: A nested list of ideals in the algebra-with-one a.

This command computes the powers of the augmentation ideal of a, and returns their list. The list stops when the list becomes stationary.

The optional second argument gives a limit to the number of terms to put in the series.

    gap> a := ThinnedAlgebraWithOne(GF(2),GrigorchukGroup);
    <self-similar algebra-with-one on alphabet GF(2)^2 with 4 generators>
    gap> q := MatrixQuotient(a,3);
    <algebra-with-one of dimension 22 over GF(2)>
    gap> l := DimensionSeries(q);
    [ <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (5 generators)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 21)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 18)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 14)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 10)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 6)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 3)>,
      <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 1)>,
      <algebra of dimension 0 over GF(2)> ]

##### 11.8-12 AlgebraHomomorphismByFunction

`‣ AlgebraHomomorphismByFunction`( A, B, f ) | ( operation )

`‣ AlgebraWithOneHomomorphismByFunction`( A, B, f ) | ( operation )

Returns: A homomorphism from the algebra A to the algebra B.

These functions construct an algebra homomorphism from a one-argument function. They do not check that the function actually defines a homomorphism.

    gap> A := MatrixAlgebra(Rationals,2);
    ( Rationals^[ 2, 2 ] )
    gap> e1 := AlgebraHomomorphismByFunction(Rationals,A,f->[[f,0],[0,0]]);
    MappingByFunction( Rationals, ( Rationals^[ 2, 2 ] ), function( f ) ... end )
    gap> 11^e1;
    [ [ 11, 0 ], [ 0, 0 ] ]

##### 11.8-13 IsFpLieAlgebra

`‣ IsFpLieAlgebra` | ( filter )

The category of Lie algebras coming from a finitely presented group. They appear as the `JenningsLieAlgebra` (Reference: JenningsLieAlgebra) of a finitely presented group.

If `G` is an infinite, finitely presented group, then the original implementation of `JenningsLieAlgebra` (Reference: JenningsLieAlgebra) does not return. On the other hand, the implementation in FR constructs a graded object, for which the graded components are computed on-demand; see `JenningsLieAlgebra` (11.8-14).

##### 11.8-14 JenningsLieAlgebra

`‣ JenningsLieAlgebra`( ring, fpgroup ) | ( operation )

Returns: The Jennings Lie algebra of fpgroup.

This method does not compute the Jennings Lie algebra per se; it merely constructs a placeholder to contain the result.

    gap> f := FreeGroup(4);
    <free group on the generators [ f1, f2, f3, f4 ]>
    gap> surfacegp := f/[Comm(f.1,f.2)*Comm(f.3,f.4)];
    <fp group of size infinity on the generators [ f1, f2, f3, f4 ]>
    gap> j := JenningsLieAlgebra(Rationals,surfgp);
    <FP Lie algebra over Rationals>
    gap> List([1..4],Grading(j).hom_components);
    [ <vector space over Rationals, with 4 generators>,
      <vector space over Rationals, with 5 generators>,
      <vector space over Rationals, with 16 generators>,
      <vector space over Rationals, with 45 generators> ]
    gap> B := Basis(Grading(j).hom_components(1));
    gap> B[1]*B[2]+B[3]*B[4];
    <zero Lie element>

##### 11.8-15 SolutionMatModN

`‣ SolutionMatModN`( mat, vec, N ) | ( operation )

Solve the linear system `sol*mat=vec` modulo N. The arguments are assumed to be an integer matrix and vector. Either returns an integer solution, or `fail` if no such solution exists.

##### 11.8-16 SolutionMatMod1

`‣ SolutionMatMod1`( mat, vec ) | ( operation )

Solve the linear system `sol*mat=vec` in \(Q/Z\). The arguments are assumed to be rational matrices. Assuming there are finitely many solutions, returns them all.

##### 11.8-17 CyclotomicByArgument

`‣ CyclotomicByArgument`( q ) | ( operation )

Returns: The cyclotomic field element equal to \(\exp(2\pi i q)\).

##### 11.8-18 ArgumentOfCyclotomic

`‣ ArgumentOfCyclotomic`( z ) | ( operation )

Returns: The rational \(q\) such that \(\exp(2\pi i q)=z\).

#### 11.9 User settings

##### 11.9-1 InfoFR

`‣ InfoFR` | ( info class )

This is an `Info` class for the package FR. The command `SetInfoLevel(InfoFR,1);` switches on the printing of some information during the computations of certain FR functions; in particular all automatic conversions between FR machines and Mealy machines.

The command `SetInfoLevel(InfoFR,2);` requests a little more information, and in particular prints intermediate results in potentially long calculations such as `NucleusOfFRSemigroup` (7.2-19).

The command `SetInfoLevel(InfoFR,3);` ensures that FR will print information every few seconds or so. This is useful to gain confidence that the program is not stuck due to a programming bug by the author of FR.

##### 11.9-2 SEARCH@

`‣ SEARCH@` | ( global variable )

This variable controls the search mechanism in FR groups. It is a record with in particular entries `radius` and `depth`.

`radius` limits the search in FR groups to balls of that radius in the generating set. For example, the command `x in G` will initiate a search in `G` to attempt to express `x` as a reasonably short word in the generators of `G`.

`depth` limits the level of the tree on which quotients of FR groups should be considered. Again for the command `x in G`, deeper and deeper quotients will be considered, in the hope of finding a quotient of `G` to which `x` does not belong.

A primitive mechanism is implemented to search alternatively for a quotient disproving `x in G` and a word proving `x in G`.

When the limits are reached and the search was unsuccessful, an interactive `Error()` is raised, to let the user increase their values.

Specific limits can be passed to any command via the options `FRdepth` and `FRradius`, as for example in `Size(G:FRdepth:=3,FRradius:=5)`.

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