  
  [1X2 [33X[0;0YAlgebraic Properties of Braces[133X[101X
  
  
  [1X2.1 [33X[0;0YBraces and Radical Rings[133X[101X
  
  [1X2.1-1 AdditiveGroupOfRing[101X
  
  [33X[1;0Y[29X[2XAdditiveGroupOfRing[102X( [3Xring[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya group[133X
  
  [33X[0;0YThis  function returns a permutation representation of the additive group of
  the given ring.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrg := SmallRing(8,10);;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(AdditiveGroupOfRing(rg));[127X[104X
    [4X[28X"C4 x C2"[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 IsJacobsonRadical[101X
  
  [33X[1;0Y[29X[2XIsJacobsonRadical[102X( [3Xring[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ytrue if the ring is radical and false otherwise.[133X
  
  [33X[0;0YThis function checks whether a ring is Jacobson radical.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrg := SmallRing(8,11);;[127X[104X
    [4X[25Xgap>[125X [27XIsJacobsonRadical(rg);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrg := SmallRing(8,20);;[127X[104X
    [4X[25Xgap>[125X [27XIsJacobsonRadical(rg);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YBraces and Yang-Baxter Equation[133X[101X
  
  [1X2.2-1 Table2YB[101X
  
  [33X[1;0Y[29X[2XTable2YB[102X( [3Xtable[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe solution given by the table[133X
  
  [33X[0;0YGiven the table with [23Xr(x,y)[123X in the position [23X(x,y)[123X find the corresponding [23Xr[123X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xl := Table(SmallIYB(4,13));;[127X[104X
    [4X[25Xgap>[125X [27Xt := Table2YB(l);;[127X[104X
    [4X[25Xgap>[125X [27XIdCycleSet(YB2CycleSet(t));[127X[104X
    [4X[28X[ 4, 13 ][128X[104X
  [4X[32X[104X
  
  [1X2.2-2 Evaluate[101X
  
  [33X[1;0Y[29X[2XEvaluate[102X( [3Xobj[103X, [3Xpair[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya pair of two integers[133X
  
  [33X[0;0YGiven the pair [23X(x,y)[123X this function returns [23Xr(x,y)[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcs := SmallCycleSet(4,13);;[127X[104X
    [4X[25Xgap>[125X [27Xyb := CycleSet2YB(cs);;[127X[104X
    [4X[25Xgap>[125X [27XPermutations(yb);[127X[104X
    [4X[28X[ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ], [128X[104X
    [4X[28X  [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ][128X[104X
    [4X[25Xgap>[125X [27XEvaluate(yb, [1,2]);[127X[104X
    [4X[28X[ 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27XEvaluate(yb, [1,3]); [127X[104X
    [4X[28X[ 4, 2 ][128X[104X
  [4X[32X[104X
  
  [1X2.2-3 LyubashenkoYB[101X
  
  [33X[1;0Y[29X[2XLyubashenkoYB[102X( [3Xsize[103X, [3Xf[103X, [3Xg[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya permutation solution to the YBE[133X
  
  [33X[0;0YFinite  Lyubashenko  (or  permutation) solutions are defined as follows: Let
  [23XX=\{1,\dots,n\}[123X and [23Xf,g\colon X\to X[123X be bijective functions such that [23Xfg=gf[123X.
  Then  [23X(X,r)[123X,  where  [23Xr(x,y)=(f(y),g(x))[123X,  is a set-theoretic solution to the
  YBE.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xyb := LyubashenkoYB(4, (1,2),(3,4));[127X[104X
    [4X[28X<A set-theoretical solution of size 4>[128X[104X
    [4X[25Xgap>[125X [27XPermutations(last);[127X[104X
    [4X[28X[ [ (1,2), (1,2), (1,2), (1,2) ], [ (3,4), (3,4), (3,4), (3,4) ] ][128X[104X
  [4X[32X[104X
  
  [1X2.2-4 IsIndecomposable[101X
  
  [33X[1;0Y[29X[2XIsIndecomposable[102X( [3XX[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue if the involutive solutions is indecomposable[133X
  
  [1X2.2-5 Table[101X
  
  [33X[1;0Y[29X[2XTable[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya table with the image of the solution[133X
  
  [33X[0;0YThe table shows the value of [23Xr(x,y)[123X for each [23X(x,y)[123X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xyb := SmallIYB(3,2);;[127X[104X
    [4X[25Xgap>[125X [27XTable(yb);[127X[104X
    [4X[28X[ [ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ] ], [ [ 1, 2 ], [ 2, 2 ], [ 3, 1 ] ], [ [ 2, 3 ], [ 1, 3 ], [ 3, 3 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X2.2-6 DehornoyClass[101X
  
  [33X[1;0Y[29X[2XDehornoyClass[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe class of an involutive solution[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcs := SmallCycleSet(4,13);;[127X[104X
    [4X[25Xgap>[125X [27Xyb := CycleSet2YB(cs);;[127X[104X
    [4X[25Xgap>[125X [27XDehornoyClass(yb);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xcs := SmallCycleSet(4,19);;[127X[104X
    [4X[25Xgap>[125X [27Xyb := CycleSet2YB(cs);;[127X[104X
    [4X[25Xgap>[125X [27XDehornoyClass(yb);[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [1X2.2-7 DehornoyRepresentationOfStructureGroup[101X
  
  [33X[1;0Y[29X[2XDehornoyRepresentationOfStructureGroup[102X( [3Xobj[103X, [3Xvariable[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA faithful linear representaiton of the structure group of [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcs := SmallCycleSet(4,13);;[127X[104X
    [4X[25Xgap>[125X [27Xyb := CycleSet2YB(cs);;[127X[104X
    [4X[25Xgap>[125X [27XPermutations(yb);[127X[104X
    [4X[28X[ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ], [128X[104X
    [4X[28X  [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ][128X[104X
    [4X[25Xgap>[125X [27Xfield := FunctionField(Rationals, 1);;[127X[104X
    [4X[25Xgap>[125X [27Xq := IndeterminatesOfFunctionField(field)[1];;[127X[104X
    [4X[25Xgap>[125X [27XG := DehornoyRepresentationOfStructureGroup(yb, q);;[127X[104X
    [4X[25Xgap>[125X [27Xx1 := G.1;;[127X[104X
    [4X[25Xgap>[125X [27Xx2 := G.2;;[127X[104X
    [4X[25Xgap>[125X [27Xx3 := G.3;;[127X[104X
    [4X[25Xgap>[125X [27Xx4 := G.4;;[127X[104X
    [4X[25Xgap>[125X [27Xx1*x2=x2*x4;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx1*x3=x4*x2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx1*x4=x3*x3;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx2*x1=x3*x4;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx2*x2=x4*x1;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx3*x1=x4*x3;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-8 IdYB[101X
  
  [33X[1;0Y[29X[2XIdYB[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe identification number of [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcs := SmallCycleSet(5,10);;[127X[104X
    [4X[25Xgap>[125X [27XIdCycleSet(cs);[127X[104X
    [4X[28X[ 5, 10 ][128X[104X
    [4X[25Xgap>[125X [27Xcs := SmallCycleSet(4,3);;[127X[104X
    [4X[25Xgap>[125X [27Xyb := CycleSet2YB(cs);;[127X[104X
    [4X[25Xgap>[125X [27XIdYB(yb);[127X[104X
    [4X[28X[ 4, 3 ][128X[104X
  [4X[32X[104X
  
  [1X2.2-9 LinearRepresentationOfStructureGroup[101X
  
  [33X[1;0Y[29X[2XLinearRepresentationOfStructureGroup[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe  permutation  brace of the involutive solution of [3Xobj[103X a linear
            representation  of  the  structure  group  of  a finite involutive
            solution[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xyb := SmallIYB(5,86);;[127X[104X
    [4X[25Xgap>[125X [27XIdBrace(IYBBrace(yb));[127X[104X
    [4X[28X[ 6, 2 ][128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xyb := SmallIYB(5,86);;[127X[104X
    [4X[25Xgap>[125X [27Xgr := LinearRepresentationOfStructureGroup(yb);;[127X[104X
    [4X[25Xgap>[125X [27Xgens := GeneratorsOfGroup(gr);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(gens[1]);[127X[104X
    [4X[28X[ [  0,  1,  0,  0,  0,  1 ],[128X[104X
    [4X[28X  [  1,  0,  0,  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  0,  1,  0 ],[128X[104X
    [4X[28X  [  0,  0,  1,  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  0,  0,  1 ] ][128X[104X
  [4X[32X[104X
  
