GAP (YangBaxter) - Chapter 3: Ideals and left ideals

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3 Ideals and left ideals
3.1 Left ideals
3.1-1 LeftIdeals
3.1-2 StrongLeftIdeals
3.1-3 IsLeftIdeal
3.2 Ideals
3.2-1 IsIdeal
3.2-2 Ideals
3.2-3 AsIdeal
3.2-4 IdealGeneratedBy
3.2-5 IntersectionOfTwoIdeals
3.2-6 SumOfTwoIdeals
3.3 Sequences (left) ideals
3.3-1 LeftSeries
3.3-2 RightSeries
3.3-3 IsLeftNilpotent
3.3-4 IsSimpleSkewbrace
3.3-5 IsRightNilpotent
3.3-6 LeftNilpotentIdeals
3.3-7 RightNilpotentIdeals
3.3-8 SmoktunowiczSeries
3.3-9 Socle
3.4 Mutipermutation skew braces
3.4-1 SocleSeries
3.4-2 MultipermutationLevel
3.4-3 IsMultipermutation
3.4-4 Fix
3.4-5 KernelOfLambda
3.4-6 Quotient
3.5 Prime and semiprime ideals
3.5-1 IsPrimeBrace
3.5-2 IsPrimeIdeal
3.5-3 PrimeIdeals
3.5-4 IsSemiprime
3.5-5 IsSemiprimeIdeal
3.5-6 SemiprimeIdeals
3.5-7 BaerRadical
3.5-8 IsBaer
3.5-9 WedderburnRadical
3.5-10 SolvableSeries
3.5-11 IsMinimalIdeal
3.5-12 MinimalIdeals

### 3 Ideals and left ideals

In this section we describe several functions related to ideals and left ideals of skew braces. References: [GV17] and [SV18].

#### 3.1 Left ideals

An left ideal \(I\) of a skew brace \(A\) is a subgroup \(I\) of the additive group of \(A\) such that \(\lambda_a(I)\subseteq I\) for all \(a\in A\).

##### 3.1-1 LeftIdeals

`‣ LeftIdeals`( obj ) | ( attribute )

Returns: a list with the left ideals of the skew brace obj

##### 3.1-2 StrongLeftIdeals

`‣ StrongLeftIdeals`( obj ) | ( attribute )

Returns: a list with the left ideals of the skew brace obj that are normal in the additive group of \(A\)

##### 3.1-3 IsLeftIdeal

`‣ IsLeftIdeal`( obj ) | ( operation )

Returns: true if the subset is a left ideal of obj

    gap> br := SmallBrace(8,4);
    <brace of size 8>
    gap> leftideals := LeftIdeals(br);
    [ <brace of size 1>, <brace of size 2>, <brace of size 4>, <brace of size 8> ]
    gap> List(leftideals, x->IsLeftIdeal(br, x));
    [ true, true, true, true ]
    gap> List(leftideals, IdBrace);
    [ [ 1, 1 ], [ 2, 1 ], [ 4, 1 ], [ 8, 4 ] ]

#### 3.2 Ideals

An ideal \(I\) of a skew brace \(A\) is a normal subgroup \(I\) of the additive group of \(A\) such that \(\lambda_a(I)\subseteq I\) and \(a\circ I=I\circ a\) for all \(a\in A\).

##### 3.2-1 IsIdeal

`‣ IsIdeal`( obj, subset ) | ( operation )

Returns: true if the subset is a left ideal of obj

    gap> br := SmallBrace(8,4);
    <brace of size 8>
    gap> leftideals := LeftIdeals(br);
    [ <brace of size 1>, <brace of size 2>, <brace of size 4>, <brace of size 8> ]
    gap> List(leftideals, x->IsLeftIdeal(br, x));
    [ true, true, true, true ]
    gap> List(leftideals, IdBrace);
    [ [ 1, 1 ], [ 2, 1 ], [ 4, 1 ], [ 8, 4 ] ]

##### 3.2-2 Ideals

`‣ Ideals`( obj ) | ( attribute )

Returns: a list with the ideals of the skew brace obj

##### 3.2-3 AsIdeal

`‣ AsIdeal`( arg1, arg2 ) | ( operation )

##### 3.2-4 IdealGeneratedBy

`‣ IdealGeneratedBy`( obj, subset ) | ( operation )

Returns: the ideal of obj generated by the given subset

The ideal of a skew brace \(A\) generated by a subset \(X\) is the intersection of all the ideals of \(A\) containing \(X\).

    gap> br := SmallSkewbrace(6,6);;
    gap> AsList(br);
    [ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)>, <(1,4)(2,5)(3,6)>,
      <(1,5,3,4,2,6)>, <(1,6,2,4,3,5)> ]
    gap> IdealGeneratedBy(br, [last[2]]);
    <brace of size 3>

##### 3.2-5 IntersectionOfTwoIdeals

`‣ IntersectionOfTwoIdeals`( ideal1, ideal2 ) | ( operation )

Returns: the intersection of ideal1 and ideal2

    gap> br := SmallSkewbrace(6,6);;
    gap> Ideals(br);
    [ <brace of size 6>, <brace of size 2>, <brace of size 3>, <brace of size 1> ]
    gap> IntersectionOfTwoIdeals(last[2],last[3]);
    <brace of size 1>

##### 3.2-6 SumOfTwoIdeals

`‣ SumOfTwoIdeals`( ideal1, ideal2 ) | ( operation )

Returns: the sum of ideal1 and ideal2

    gap> br := SmallSkewbrace(6,6);;
    gap> Ideals(br);
    [ <brace of size 6>, <brace of size 2>, <brace of size 3>, <brace of size 1> ]
    gap> SumOfTwoIdeals(last[2],last[3]);
    <brace of size 6>

#### 3.3 Sequences (left) ideals

##### 3.3-1 LeftSeries

`‣ LeftSeries`( obj ) | ( attribute )

Returns: the left ideals of the left series of obj

The left series of a skew brace \(A\) is defined recursively as \(A^1=A\) and \(A^{n+1}=A*A^n\) for \(n\geq1\), where \(a*b=\lambda_a(b)-b\). Each \(A^n\) is a left ideal.

    gap> br := SmallSkewbrace(8,20);
    <skew brace of size 8>
    gap> LeftSeries(br);
    [ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ]

##### 3.3-2 RightSeries

`‣ RightSeries`( obj ) | ( attribute )

Returns: the ideals of the right series of obj

The right series of a skew brace 0\(A\) is defined recursively as \(A^{(1)}=A\) and \(A^{(n+1)}=A*A^{(n)}\) for \(n\geq1\), where \(a*b=\lambda_a(b)-b\)

    gap> br := SmallSkewbrace(8,20);
    <skew brace of size 8>
    gap> RightSeries(br);
    [ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ]

##### 3.3-3 IsLeftNilpotent

`‣ IsLeftNilpotent`( obj ) | ( property )

Returns: true if the skew brace obj is left nilpotent.

A skew brace \(A\) is said to be left nilpotent if there exists \(n\geq1\) such that \(A^n=0\).

    gap> IsLeftNilpotent(SmallBrace(8,18));
    true
    gap> IsLeftNilpotent(SmallBrace(12,2));
    false

##### 3.3-4 IsSimpleSkewbrace

`‣ IsSimpleSkewbrace`( obj ) | ( property )

Returns: true if the skew brace obj is simple.

A skew brace \(A\) is said to be simple if \(\{0\}\) and \(A\) are its only ideals.

    gap> IsSimple(SmallSkewbrace(12,22));
    true
    gap> IsSimple(SmallSkewbrace(12,21));
    false

##### 3.3-5 IsRightNilpotent

`‣ IsRightNilpotent`( obj ) | ( property )

Returns: true if the skew brace obj is right nilpotent.

A skew brace \(A\) is said to be right nilpotent if there exists \(n\geq1\) such that \(A^{(n)}=0\).

    gap> IsRightNilpotent(SmallBrace(8,18));
    false
    gap> IsRightNilpotent(SmallBrace(12,2));
    true

##### 3.3-6 LeftNilpotentIdeals

`‣ LeftNilpotentIdeals`( obj ) | ( attribute )

Returns: the list of right or left nilpotent ideals of obj

An ideal \(I\) of a skew brace \(A\) is said to be left if it is left nilpotent as a skew brace.

##### 3.3-7 RightNilpotentIdeals

`‣ RightNilpotentIdeals`( obj ) | ( attribute )

Returns: the list of right or left nilpotent ideals of obj

An ideal \(I\) of a skew brace \(A\) is said to be right nilpotent if An ideal \(I\) of a skew brace \(A\) is said to be left if it is right nilpotent as a skew brace.

    gap> br := SmallBrace(8,18);;
    gap> IsLeftNilpotent(br);
    true
    gap> LeftNilpotentIdeals(br);
    [ <brace of size 8>, <brace of size 4>, <brace of size 1> ]
    gap> IsRightNilpotent(br);
    false
    gap> RightNilpotentIdeals(br);
    [ <brace of size 4>, <brace of size 1> ]

##### 3.3-8 SmoktunowiczSeries

`‣ SmoktunowiczSeries`( obj, bound ) | ( operation )

Returns: a list of bound left ideals of the Smoktunowicz's series of obj

The Smoktunowicz's series of a skew brace \(A\) is defined recursively as \(A^{[1]}=A\) and \(A^{[n+1]}\) is the additive subgroup of \(A\) generated by \(A^{[i]}*A^{[n+1-i]}\) for \(1\leq i+j\leq n+1\), where \(a*b=\lambda_a(b)-b\).

    gap> br := SmallBrace(16,145);;
    gap> SmoktunowiczSeries(br,4);
    [ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,
      <brace of size 2> ]
    gap> SmoktunowiczSeries(br,5);
    [ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,
      <brace of size 2>, <brace of size 1> ]

##### 3.3-9 Socle

`‣ Socle`( obj ) | ( attribute )

Returns: the ideals of the socle series of obj

The socle of a skew brace \(A\) is the ideal \(\ker\lambda\cap Z(A,+)\).

    gap> Socle(SmallSkewbrace(6,2));
    <brace of size 1>
    gap> Socle(SmallBrace(8,20));
    <brace of size 8>
    gap> Socle(SmallBrace(8,2));
    <brace of size 4>

#### 3.4 Mutipermutation skew braces

##### 3.4-1 SocleSeries

`‣ SocleSeries`( obj ) | ( operation )

Returns: the socle series of obj

The socle series of a skew brace \(A\) is defined recursively as \(A_1=A\) and \(A_{n+1}=A_n/\mathrm{Soc}(A_n)\), see [SV18].

##### 3.4-2 MultipermutationLevel

`‣ MultipermutationLevel`( obj ) | ( attribute )

Returns: the multipermutation level of the skew brace obj

The multipermutation level of a skew brace \(A\) is defined as the smallest positive integer \(n\) such that the \(n\)-th term \(A_n\) of the socle series has only one element, see Definition 5.17 of [SV18].

    gap> br := SmallBrace(8,20);;
    gap> SocleSeries(br);
    [ <brace of size 8>, <brace of size 1> ]
    gap> MultipermutationLevel(br);
    2

##### 3.4-3 IsMultipermutation

`‣ IsMultipermutation`( obj ) | ( property )

Returns: true if the skew brace obj has finite multipermutation level and false otherwise

##### 3.4-4 Fix

`‣ Fix`( obj ) | ( attribute )

Returns: true if the skew brace obj has finite multipermutation level and false otherwise

##### 3.4-5 KernelOfLambda

`‣ KernelOfLambda`( obj ) | ( attribute )

Returns: the kernel of the map \(\lambda\) as a subset of elements of the skew brace obj.

    gap> br := SmallBrace(6,1);;
    gap> KernelOfLambda(br);
    [ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)> ]

##### 3.4-6 Quotient

`‣ Quotient`( obj, ideal ) | ( operation )

Returns: the quotient obj by ideal

    gap> br := SmallBrace(8,10);;
    gap> ideals := Ideals(br);;
    gap> Quotient(br, ideals[5]);
    <brace of size 4>
    gap> br/ideals[2];
    <brace of size 2>

#### 3.5 Prime and semiprime ideals

##### 3.5-1 IsPrimeBrace

`‣ IsPrimeBrace`( obj ) | ( property )

Returns: true if the skew brace obj is prime

A skew brace \(A\) is said to be prime if for all non-zero ideals \(I\) and \(J\) one has \(I*J\ne 0\)

    gap> IsPrimeBrace(SmallBrace(24,12));
    false
    gap> IsPrimeBrace(SmallBrace(24,94));
    true

##### 3.5-2 IsPrimeIdeal

`‣ IsPrimeIdeal`( obj ) | ( property )

Returns: true if the ideal obj is prime

An ideal \(I\) of a skew brace \(A\) is said to be prime if \(A/I\) is a prime skew brace.

    gap> br := SmallBrace(24,94);
    <brace of size 24>
    gap> IsPrimeBrace(br);
    true
    gap> Ideals(br);
    [ <brace of size 24>, <brace of size 1> ]
    gap> IsPrimeIdeal(last[2]);
    true

##### 3.5-3 PrimeIdeals

`‣ PrimeIdeals`( obj ) | ( attribute )

Returns: the list of prime ideals of the skew brace obj

    gap> PrimeIdeals(SmallBrace(24,94));
    [ <brace of size 24>, <brace of size 1> ]

##### 3.5-4 IsSemiprime

`‣ IsSemiprime`( obj ) | ( attribute )

Returns: true if the skew brace obj is semiprime

An ideal \(I\) of a skew brace \(A\) is said to be semiprime if \(A/I\) is a semiprime skew brace.

    gap> br := DirectProductSkewbraces(SmallSkewbrace(12,22),SmallSkewbrace(12,22));;
    gap> IsSemiprime(br);
    true

##### 3.5-5 IsSemiprimeIdeal

`‣ IsSemiprimeIdeal`( obj ) | ( attribute )

Returns: true if the ideal obj is semiprime

    gap> SemiprimeIdeals(SmallSkewbrace(12,24));
    [ <skew brace of size 12> ]
    gap> IsSemiprimeIdeal(last[1]);
    true

##### 3.5-6 SemiprimeIdeals

`‣ SemiprimeIdeals`( obj ) | ( attribute )

Returns: the list of semiprime ideals of the skew brace obj

    gap> SemiprimeIdeals(SmallSkewbrace(12,22));
    [ <skew brace of size 12>, <brace of size 1> ]
    gap> SemiprimeIdeals(SmallSkewbrace(12,24));
    [ <skew brace of size 12> ]

##### 3.5-7 BaerRadical

`‣ BaerRadical`( obj ) | ( attribute )

Returns: the Baer radical of the skew brace obj

    gap> br := SmallSkewbrace(6,2);;
    gap> BaerRadical(br);
    <skew brace of size 6>

##### 3.5-8 IsBaer

`‣ IsBaer`( obj ) | ( property )

Returns: true if the skew brace obj is ia Baer radical skew brace.

A skew brace \(A\) is said to be Baer radical if \(A=B(A)\), where \(B(A)\) is the Baer radical of \(A\) (i.e., the intersection of all prime ideals of \(A\)).

    gap> br := SmallSkewbrace(6,2);;
    gap> IsBaer(br);
    true

##### 3.5-9 WedderburnRadical

`‣ WedderburnRadical`( obj ) | ( attribute )

Returns: the Wedderburn radical of the skew brace obj

The Wedderburn radical of a skew brace is the intersection of all its prime ideals

    gap> br := SmallSkewbrace(6,2);;
    gap> WedderburnRadical(br);
    <brace of size 3>

##### 3.5-10 SolvableSeries

`‣ SolvableSeries`( obj ) | ( attribute )

Returns: a list with the solvable series of the skew brace obj

The solvable series of a skew brace \(A\) is defined recursively as \(A_{1}=A\) and \(A_{n+1}=A_{n}*A_{n}\) for \(n\geq1\), where \(a*b=\lambda_a(b)-b\)

    gap> br := SmallSkewbrace(8,20);;
    gap> IsSolvable(br);
    true
    gap> SolvableSeries(br);
    [ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ]
    gap> br := SmallSkewbrace(12,23);;
    gap> IsSolvable(br);
    false

##### 3.5-11 IsMinimalIdeal

`‣ IsMinimalIdeal`( obj, ideal ) | ( property )

Returns: true if ideal is a minimal ideal of obj An ideal \(I\) of \(A\) is said to be minimal if does not contain any other ideal of \(A\). To check if an ideal \(I\) of \(A\) is minimal, one computes the ideals of \(I\) and keep only those that are simple as a skew brace.

##### 3.5-12 MinimalIdeals

`‣ MinimalIdeals`( obj ) | ( attribute )

Returns: a list of minimal ideals of the skew brace obj

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