In this section we describe several functions related to ideals and left ideals of skew braces. References: [GV17] and [SV18].
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 \(a\in A\).
‣ LeftIdeals( obj ) | ( attribute ) |
Returns: a list with the left ideals of the skew brace obj
‣ StrongLeftIdeals( obj ) | ( attribute ) |
Returns: a list with the left ideals of the skew brace obj that are normal in the additive group of \(A\)
‣ 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 ] ]
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 \(a\in A\).
‣ 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 ] ]
‣ Ideals( obj ) | ( attribute ) |
Returns: a list with the ideals of the skew brace obj
‣ AsIdeal( arg1, arg2 ) | ( operation ) |
‣ 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>
‣ 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]); <brace of size 1>
‣ 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]); <brace of size 6>
‣ 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> ]
‣ 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> ]
‣ 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
‣ 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
‣ 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
‣ 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.
‣ 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
‣ 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> ]
‣ 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)); <brace of size 1> gap> Socle(SmallBrace(8,20)); <brace of size 8> gap> Socle(SmallBrace(8,2)); <brace of size 4>
‣ 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)); <brace of size 2> gap> Annihilator(SmallSkewbrace(4,2)); <brace of size 2> gap> Annihilator(SmallSkewbrace(8,14)); <brace of size 4>
‣ 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].
‣ 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
‣ IsMultipermutation( obj ) | ( property ) |
Returns: true if the skew brace obj has finite multipermutation level and false otherwise
‣ 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)> ]
‣ 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)> ]
‣ 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>
‣ 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
‣ 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
‣ PrimeIdeals( obj ) | ( attribute ) |
Returns: the list of prime ideals of the skew brace obj
gap> Length(PrimeIdeals(SmallBrace(24,94))); 2
‣ 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
‣ 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
‣ SemiprimeIdeals( obj ) | ( attribute ) |
Returns: the list of semiprime ideals of the skew brace obj
gap> SemiprimeIdeals(SmallSkewbrace(12,24)); [ <skew brace of size 12> ] gap> Length(SemiprimeIdeals(SmallSkewbrace(12,22))); 2
‣ 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>
‣ 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
‣ 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>
‣ 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
‣ 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.
‣ MinimalIdeals( obj ) | ( attribute ) |
Returns: a list of minimal ideals of the skew brace obj
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