Algebraic Properties of Braces

Braces and Radical Rings a group This function returns a permutation representation of the additive group of the given ring.
gap> rg := SmallRing(8,10);; gap> StructureDescription(AdditiveGroupOfRing(rg)); "C4 x C2"
true if the ring is radical and false otherwise. This function checks whether a ring is Jacobson radical.
gap> rg := SmallRing(8,11);; gap> IsJacobsonRadical(rg); true gap> rg := SmallRing(8,20);; gap> IsJacobsonRadical(rg); false
Braces and Yang-Baxter Equation a pair of two integers Given the pair (x,y) this function returns r(x,y).
gap> cs := SmallCycleSet(4,13);; gap> yb := CycleSet2YB(cs);; gap> Permutations(yb); [ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ], [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ] gap> Evaluate(yb, [1,2]); [ 2, 4 ] gap> Evaluate(yb, [1,3]); [ 4, 2 ]
a permutation solution to the YBE Finite Lyubashenko (or permutation) solutions are defined as follows: Let X=\{1,\dots,n\} and f,g\colon X\to X be bijective functions such that fg=gf. Then (X,r), where r(x,y)=(f(y),g(x)), is a set-theoretic solution to the YBE.
gap> yb := LyubashenkoYB(4, (1,2),(3,4)); <A set-theoretical solution of size 4> gap> Permutations(last); [ [ (1,2), (1,2), (1,2), (1,2) ], [ (3,4), (3,4), (3,4), (3,4) ] ]
The class of an involutive solution
gap> cs := SmallCycleSet(4,13);; gap> yb := CycleSet2YB(cs);; gap> DehornoyClass(yb); 2 gap> cs := SmallCycleSet(4,19);; gap> yb := CycleSet2YB(cs);; gap> DehornoyClass(yb); 4
A faithful linear representaiton of the structure group of obj

gap> cs := SmallCycleSet(4,13);; gap> yb := CycleSet2YB(cs);; gap> Permutations(yb); [ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ], [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ] gap> field := FunctionField(Rationals, 1);; gap> q := IndeterminatesOfFunctionField(field)[1];; gap> G := DehornoyRepresentationOfStructureGroup(yb, q);; gap> x1 := G.1;; gap> x2 := G.2;; gap> x3 := G.3;; gap> x4 := G.4;; gap> x1*x2=x2*x4; true gap> x1*x3=x4*x2; true gap> x1*x4=x3*x3; true gap> x2*x1=x3*x4; true gap> x2*x2=x4*x1; true gap> x3*x1=x4*x3; true
the identification number of obj

gap> cs := SmallCycleSet(5,10);; gap> IdCycleSet(cs); [ 5, 10 ] gap> cs := SmallCycleSet(4,3);; gap> yb := CycleSet2YB(cs);; gap> IdYB(yb); [ 4, 3 ]
the permutation brace of the involutive solution of obj a linear representation of the structure group of a finite involutive solution

gap> yb := SmallIYB(5,86);; gap> IdBrace(IYBBrace(yb)); [ 6, 2 ]

gap> yb := SmallIYB(5,86);; gap> gr := LinearRepresentationOfStructureGroup(yb);; gap> gens := GeneratorsOfGroup(gr);; gap> Display(gens[1]); [ [ 0, 1, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ] ]