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
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)> ]
‣ 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 ] ]
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); [ <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 ] ]
‣ 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]]); <ideal in <brace of size 6>, (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]); <ideal in <brace of size 6>, (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]); <ideal in <brace of size 6>, (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>, <left ideal in <skew brace of size 8>, (size 2)>, <left ideal in <skew brace of size 8>, (size 1)> ]
‣ 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)> ]
‣ 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)); <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)>
‣ 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)>
‣ 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)); [ <ideal in <skew brace of size 12>, (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)); [ <ideal in <skew brace of size 12>, (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); <ideal in <skew brace of size 6>, (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); <ideal in <skew brace of size 6>, (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
generated by GAPDoc2HTML