  
  [1X5 [33X[0;0YBasic Methods And Attributes[133X[101X
  
  [33X[0;0YIn  this  chapter  we  describe the basic core methods and attributes of the
  [5XLOOPS[105X package.[133X
  
  
  [1X5.1 [33X[0;0YBasic Attributes[133X[101X
  
  [33X[0;0YWe  associate  many  attributes  with  quasigroups  in  order  to  speed  up
  computation.  This  section  lists  some basic attributes of quasigroups and
  loops.[133X
  
  [1X5.1-1 Elements[101X
  
  [33X[1;0Y[29X[2XElements[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe list of elements of a quasigroup [3XQ[103X.[133X
  
  [33X[0;0YSee Section [14X3.4[114X for more information about element labels.[133X
  
  [1X5.1-2 CayleyTable[101X
  
  [33X[1;0Y[29X[2XCayleyTable[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe Cayley table of a quasigroup [3XQ[103X.[133X
  
  [33X[0;0YSee Section [14X4.1[114X for more information about quasigroup Cayley tables.[133X
  
  [1X5.1-3 One[101X
  
  [33X[1;0Y[29X[2XOne[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe identity element of a loop [3XQ[103X.[133X
  
  [33X[0;0Y[12XRemark:[112XIf  you want to know if a quasigroup [3XQ[103X has a neutral element, you can
  find      out      with     the     standard     function     for     magmas
  [10XMultiplicativeNeutralElement([3XQ[103X[10X)[110X.[133X
  
  [1X5.1-4 Size[101X
  
  [33X[1;0Y[29X[2XSize[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe size of a quasigroup [3XQ[103X.[133X
  
  [1X5.1-5 Exponent[101X
  
  [33X[1;0Y[29X[2XExponent[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe  exponent  of a power associative loop [3XQ[103X. (The method does not
            test if [3XQ[103X is power associative.)[133X
  
  [33X[0;0YWhen  [3XQ[103X  is  a  [13Xpower  associative loop[113X, that is, the powers of elements are
  well-defined  in  [3XQ[103X, then the [13Xexponent[113X of [3XQ[103X is the smallest positive integer
  divisible by the orders of all elements of [3XQ[103X.[133X
  
  
  [1X5.2 [33X[0;0YBasic Arithmetic Operations[133X[101X
  
  [33X[0;0YEach  quasigroup  element in [5XGAP[105X knows to which quasigroup it belongs. It is
  therefore possible to perform arithmetic operations with quasigroup elements
  without   referring   to  the  quasigroup.  All  elements  involved  in  the
  calculation must belong to the same quasigroup.[133X
  
  [33X[0;0YTwo elements [22Xx[122X, [22Xy[122X of the same quasigroup are multiplied by [22Xx*y[122X in [5XGAP[105X. Since
  multiplication of at least three elements is ambiguous in the nonassociative
  case,  we  parenthesize  elements by default from left to right, i.e., [22Xx*y*z[122X
  means  [22X((x*y)*z)[122X. Of course, one can specify the order of multiplications by
  providing parentheses.[133X
  
  
  [1X5.2-1 [33X[0;0YLeftDivision and RightDivision[133X[101X
  
  [33X[1;0Y[29X[2XLeftDivision[102X( [3Xx[103X, [3Xy[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRightDivision[102X( [3Xx[103X, [3Xy[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe  left  division  [3Xx[103X[22Xbackslash[122X[3Xy[103X (resp. the right division [3Xx[103X[22X/[122X[3Xy[103X) of
            two elements [3Xx[103X, [3Xy[103X of the same quasigroup.[133X
  
  [33X[1;0Y[29X[2XLeftDivision[102X( [3XS[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLeftDivision[102X( [3Xx[103X, [3XS[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRightDivision[102X( [3XS[103X, [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRightDivision[102X( [3Xx[103X, [3XS[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe   list  of  elements  obtained  by  performing  the  specified
            arithmetical  operation elementwise using a list [3XS[103X of elements and
            an element [3Xx[103X.[133X
  
  [33X[0;0Y[12XRemark:[112X  We  support  [22X/[122X  in  place  of  [10XRightDivision[110X. But we do not support
  [22Xbackslash[122X in place of [10XLeftDivision[110X.[133X
  
  
  [1X5.2-2 [33X[0;0YLeftDivisionCayleyTable and RightDivisionCayleyTable[133X[101X
  
  [33X[1;0Y[29X[2XLeftDivisionCayleyTable[102X( [3XQ[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRightDivisionCayleyTable[102X( [3XQ[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe  Cayley  table  of  the  respective  arithmetic operation of a
            quasigroup [3XQ[103X.[133X
  
  
  [1X5.3 [33X[0;0YPowers and Inverses[133X[101X
  
  [33X[0;0YPowers of elements are generally not well-defined in quasigroups. For magmas
  and  a  positive  integral  exponent, [5XGAP[105X calculates powers in the following
  way:  [22Xx^1=x[122X,  [22Xx^2k=(x^k)⋅(x^k)[122X and [22Xx^2k+1=(x^2k)⋅ x[122X. One can easily see that
  this returns [22Xx^k[122X in about [22Xlog_2(k)[122X steps. For [5XLOOPS[105X, we have decided to keep
  this  method,  but the user should be aware that the method is sound only in
  power associative quasigroups.[133X
  
  [33X[0;0YLet  [22Xx[122X  be  an  element  of  a  loop [22XQ[122X with neutral element [22X1[122X. Then the [13Xleft
  inverse[113X  [22Xx^λ[122X  of [22Xx[122X is the unique element of [22XQ[122X satisfying [22Xx^λ x=1[122X. Similarly,
  the [13Xright inverse[113X [22Xx^ρ[122X satisfies [22Xxx^ρ=1[122X. If [22Xx^λ=x^ρ[122X, we call [22Xx^-1=x^λ=x^ρ[122X the
  [13Xinverse[113X of [22Xx[122X.[133X
  
  
  [1X5.3-1 [33X[0;0YLeftInverse, RightInverse and Inverse[133X[101X
  
  [33X[1;0Y[29X[2XLeftInverse[102X( [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRightInverse[102X( [3Xx[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInverse[102X( [3Xx[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe  left inverse, right inverse and inverse, respectively, of the
            quasigroup element [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCayleyTable( Q );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5 ],[128X[104X
    [4X[28X  [ 2, 1, 4, 5, 3 ],[128X[104X
    [4X[28X  [ 3, 4, 5, 1, 2 ],[128X[104X
    [4X[28X  [ 4, 5, 2, 3, 1 ],[128X[104X
    [4X[28X  [ 5, 3, 1, 2, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xelms := Elements( Q );[127X[104X
    [4X[25Xgap>[125X [27X[ l1, l2, l3, l4, l5 ];[127X[104X
    [4X[25Xgap>[125X [27X[ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ];[127X[104X
    [4X[28X[ l5, l4, fail ][128X[104X
  [4X[32X[104X
  
  
  [1X5.4 [33X[0;0YAssociators and Commutators[133X[101X
  
  [33X[0;0YSee Section [14X2.5[114X for definitions of associators and commutators.[133X
  
  [1X5.4-1 Associator[101X
  
  [33X[1;0Y[29X[2XAssociator[102X( [3Xx[103X, [3Xy[103X, [3Xz[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe associator of the elements [3Xx[103X, [3Xy[103X, [3Xz[103X of the same quasigroup.[133X
  
  [1X5.4-2 Commutator[101X
  
  [33X[1;0Y[29X[2XCommutator[102X( [3Xx[103X, [3Xy[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe commutator of the elements [3Xx[103X, [3Xy[103X of the same quasigroup.[133X
  
  
  [1X5.5 [33X[0;0YGenerators[133X[101X
  
  
  [1X5.5-1 [33X[0;0YGeneratorsOfQuasigroup and GeneratorsOfLoop[133X[101X
  
  [33X[1;0Y[29X[2XGeneratorsOfQuasigroup[102X( [3XQ[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XGeneratorsOfLoop[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA  set of generators of a quasigroup (resp. loop) [3XQ[103X. (Both methods
            are synonyms of [10XGeneratorsOfMagma[110X.)[133X
  
  [33X[0;0YAs  usual  in  [5XGAP[105X,  one can refer to the [10Xi[110Xth generator of a quasigroup [10XQ[110X by
  [10XQ.i[110X.  Note that while it is often the case that [10X Q.i = Elements(Q)[i][110X, it is
  not necessarily so.[133X
  
  [1X5.5-2 GeneratorsSmallest[101X
  
  [33X[1;0Y[29X[2XGeneratorsSmallest[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA  generating  set  [22X{q_0[122X,  [22Xdots[122X, [22Xq_m}[122X of [3XQ[103X such that [22XQ_0=∅[122X, [22XQ_m=[122X[3XQ[103X,
            [22XQ_i=⟨ q_1[122X, [22Xdots[122X, [22Xq_i ⟩[122X, and [22Xq_i+1[122X is the least element of [3XQ[103X[22X∖ Q_i[122X.[133X
  
  [1X5.5-3 SmallGeneratingSet[101X
  
  [33X[1;0Y[29X[2XSmallGeneratingSet[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA  small generating set [22X{q_0[122X, [22Xdots[122X, [22Xq_m}[122X of [3XQ[103X obtained as follows:
            [22Xq_0[122X  is  the  least  element for which [22X⟨ q_0⟩[122X is largest possible,
            [22Xq_1[122X$ is the least element for which [22X⟨ q_0,q_1[122X is largest possible,
            and so on.[133X
  
