Ideals and left ideals In this section we describe several functions related to ideals and left ideals of skew braces. References: and .
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. a list with the left ideals of the skew brace obj

a list with the left ideals of the skew brace obj that are normal in the additive group of A

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 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. 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 ] ]
a list with the ideals of the skew brace obj

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>
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>
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>
Sequences (left) ideals 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> ]
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> ]
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
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
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
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. 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> ]
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> ]
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>
Mutipermutation skew braces 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 . 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 .
gap> br := SmallBrace(8,20);; gap> SocleSeries(br); [ <brace of size 8>, <brace of size 1> ] gap> MultipermutationLevel(br); 2
true if the skew brace obj has finite multipermutation level and false otherwise

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

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)> ]
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>
Prime and semiprime ideals 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
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
the list of prime ideals of the skew brace obj
gap> PrimeIdeals(SmallBrace(24,94)); [ <brace of size 24>, <brace of size 1> ]
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

true if the ideal obj is semiprime
gap> SemiprimeIdeals(SmallSkewbrace(12,24)); [ <skew brace of size 12> ] gap> IsSemiprimeIdeal(last[1]); true

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> ]
the Baer radical of the skew brace obj
gap> br := SmallSkewbrace(6,2);; gap> BaerRadical(br); <skew brace of size 6>
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
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>
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
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.

a list of minimal ideals of the skew brace obj