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 0A 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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