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4 Ideals and left ideals
 4.1 Left ideals
 4.2 Ideals
 4.3 Sequences (left) ideals
 4.4 Mutipermutation skew braces
 4.5 Prime and semiprime ideals

4 Ideals and left ideals

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

4.1 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.

4.1-1 LeftIdeals
‣ LeftIdeals( obj )( attribute )

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

4.1-2 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

gap> br := SmallSkewbrace(24,12);
<skew brace of size 24>
gap> strong_left_ideals := StrongLeftIdeals(br);
[ <left ideal in <skew brace of size 24>, (size 24)>,
  <left ideal in <skew brace of size 24>, (size 12)>,
  <left ideal in <skew brace of size 24>, (size 6)>,
  <left ideal in <skew brace of size 24>, (size 4)>,
  <left ideal in <skew brace of size 24>, (size 2)>,
  <left ideal in <skew brace of size 24>, (size 3)>,
  <left ideal in <skew brace of size 24>, (size 1)> ]

4.1-3 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);
[ <left ideal in <brace of size 8>, (size 1)>, <left ideal in <brace of size 8>, (size 2)>, 
<left ideal in <brace of size 8>, (size 4)>, 
<left ideal in <brace of size 8>, (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 ] ]

4.2 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.

4.2-1 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);
[ <left ideal in <brace of size 8>, (size 1)>, 
<left ideal in <brace of size 8>, (size 2)>,
<left ideal in <brace of size 8>, (size 4)>, 
<left ideal in <brace of size 8>, (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 ] ]

4.2-2 Ideals
‣ Ideals( obj )( attribute )

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

4.2-3 AsIdeal
‣ AsIdeal( arg1, arg2 )( operation )

4.2-4 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]]);
<ideal in <brace of size 6>, (size 3)>

4.2-5 IntersectionOfTwoIdeals
‣ IntersectionOfTwoIdeals( ideal1, ideal2 )( operation )

Returns: the intersection of ideal1 and ideal2

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

4.2-6 SumOfTwoIdeals
‣ SumOfTwoIdeals( ideal1, ideal2 )( operation )

Returns: the sum of ideal1 and ideal2

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

4.3 Sequences (left) ideals

4.3-1 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>, 
<left ideal in <skew brace of size 8>, (size 2)>, 
<left ideal in <skew brace of size 8>, (size 1)> ]

4.3-2 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);
[ <ideal in <skew brace of size 8>, (size 8)>, 
<ideal in <skew brace of size 8>, (size 2)>, 
<ideal in <skew brace of size 8>, (size 1)> ]

4.3-3 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

4.3-4 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

4.3-5 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

4.3-6 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.

4.3-7 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> IsRightNilpotent(br);
false
gap> Length(LeftNilpotentIdeals(br));
3
gap> Length(RightNilpotentIdeals(br));
2

4.3-8 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> ]

4.3-9 Socle
‣ Socle( obj )( attribute )

Returns: the socle of obj

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

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

4.3-10 Annihilator
‣ Annihilator( obj )( attribute )

Returns: the annihilator of obj

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

gap> Annihilator(SmallSkewbrace(8,12));
<ideal in <brace of size 8>, (size 2)>
gap> Annihilator(SmallSkewbrace(4,2));
<ideal in <skew brace of size 4>, (size 2)>
gap> Annihilator(SmallSkewbrace(8,14));
<ideal in <brace of size 8>, (size 4)>

4.4 Mutipermutation skew braces

4.4-1 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].

4.4-2 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

4.4-3 IsMultipermutation
‣ IsMultipermutation( obj )( property )

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

4.4-4 Fix
‣ Fix( obj )( attribute )

Returns: the left ideal \{x\in A:\lambda_a(x)=x\;\forall a\in A\} of the skew brace A.

gap> br := SmallSkewbrace(6,1);;
gap> IsTrivialSkewbrace(br);
true
gap> Fix(br);
[ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)>, <(1,4)(2,6)(3,5)>,
  <(1,5)(2,4)(3,6)>, <(1,6)(2,5)(3,4)> ]

4.4-5 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)> ]

4.4-6 Quotient
‣ Quotient( obj, ideal )( operation )

Returns: the quotient obj by ideal

gap> br := SmallBrace(8,10);;
gap> ideals := Ideals(br);;
gap> Quotient(br, ideals[3]);
<brace of size 4>
gap> br/ideals[3];
<brace of size 4>

4.5 Prime and semiprime ideals

4.5-1 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

4.5-2 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);;
gap> IsPrimeIdeal(last[2]);
true

4.5-3 PrimeIdeals
‣ PrimeIdeals( obj )( attribute )

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

gap> Length(PrimeIdeals(SmallBrace(24,94)));
2

4.5-4 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

4.5-5 IsSemiprimeIdeal
‣ IsSemiprimeIdeal( obj )( attribute )

Returns: true if the ideal obj is semiprime

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

4.5-6 SemiprimeIdeals
‣ SemiprimeIdeals( obj )( attribute )

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

gap> SemiprimeIdeals(SmallSkewbrace(12,24));
[ <ideal in <skew brace of size 12>, (size 12)> ]
gap> Length(SemiprimeIdeals(SmallSkewbrace(12,22)));
2

4.5-7 BaerRadical
‣ BaerRadical( obj )( attribute )

Returns: the Baer radical of the skew brace obj

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

4.5-8 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

4.5-9 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);
<ideal in <skew brace of size 6>, (size 3)>

4.5-10 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

4.5-11 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.

4.5-12 MinimalIdeals
‣ MinimalIdeals( obj )( attribute )

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

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