loops/doc/chap5.txt

  
  5 Basic Methods And Attributes
  
  In  this  chapter  we  describe the basic core methods and attributes of the
  LOOPS package.
  
  
  5.1 Basic Attributes
  
  We  associate  many  attributes  with  quasigroups  in  order  to  speed  up
  computation.  This  section  lists  some basic attributes of quasigroups and
  loops.
  
  5.1-1 Elements
  
  Elements( Q )  attribute
  Returns:  The list of elements of a quasigroup Q.
  
  See Section 3.4 for more information about element labels.
  
  5.1-2 CayleyTable
  
  CayleyTable( Q )  attribute
  Returns:  The Cayley table of a quasigroup Q.
  
  See Section 4.1 for more information about quasigroup Cayley tables.
  
  5.1-3 One
  
  One( Q )  attribute
  Returns:  The identity element of a loop Q.
  
  Remark:If  you want to know if a quasigroup Q has a neutral element, you can
  find      out      with     the     standard     function     for     magmas
  MultiplicativeNeutralElement(Q).
  
  5.1-4 Size
  
  Size( Q )  attribute
  Returns:  The size of a quasigroup Q.
  
  5.1-5 Exponent
  
  Exponent( Q )  attribute
  Returns:  The  exponent  of a power associative loop Q. (The method does not
            test if Q is power associative.)
  
  When  Q  is  a  power  associative loop, that is, the powers of elements are
  well-defined  in  Q, then the exponent of Q is the smallest positive integer
  divisible by the orders of all elements of Q.
  
  
  5.2 Basic Arithmetic Operations
  
  Each  quasigroup  element in GAP 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.
  
  Two elements x, y of the same quasigroup are multiplied by x*y in GAP. Since
  multiplication of at least three elements is ambiguous in the nonassociative
  case,  we  parenthesize  elements by default from left to right, i.e., x*y*z
  means  ((x*y)*z). Of course, one can specify the order of multiplications by
  providing parentheses.
  
  
  5.2-1 LeftDivision and RightDivision
  
  LeftDivision( x, y )  operation
  RightDivision( x, y )  operation
  Returns:  The  left  division  xbackslashy (resp. the right division x/y) of
            two elements x, y of the same quasigroup.
  
  LeftDivision( S, x )  operation
  LeftDivision( x, S )  operation
  RightDivision( S, x )  operation
  RightDivision( x, S )  operation
  Returns:  The   list  of  elements  obtained  by  performing  the  specified
            arithmetical  operation elementwise using a list S of elements and
            an element x.
  
  Remark:  We  support  /  in  place  of  RightDivision. But we do not support
  backslash in place of LeftDivision.
  
  
  5.2-2 LeftDivisionCayleyTable and RightDivisionCayleyTable
  
  LeftDivisionCayleyTable( Q )  operation
  RightDivisionCayleyTable( Q )  operation
  Returns:  The  Cayley  table  of  the  respective  arithmetic operation of a
            quasigroup Q.
  
  
  5.3 Powers and Inverses
  
  Powers of elements are generally not well-defined in quasigroups. For magmas
  and  a  positive  integral  exponent, GAP calculates powers in the following
  way:  x^1=x,  x^2k=(x^k)⋅(x^k) and x^2k+1=(x^2k)⋅ x. One can easily see that
  this returns x^k in about log_2(k) steps. For LOOPS, we have decided to keep
  this  method,  but the user should be aware that the method is sound only in
  power associative quasigroups.
  
  Let  x  be  an  element  of  a  loop Q with neutral element 1. Then the left
  inverse  x^λ  of x is the unique element of Q satisfying x^λ x=1. Similarly,
  the right inverse x^ρ satisfies xx^ρ=1. If x^λ=x^ρ, we call x^-1=x^λ=x^ρ the
  inverse of x.
  
  
  5.3-1 LeftInverse, RightInverse and Inverse
  
  LeftInverse( x )  operation
  RightInverse( x )  operation
  Inverse( x )  operation
  Returns:  The  left inverse, right inverse and inverse, respectively, of the
            quasigroup element x.
  
    Example  
    gap> CayleyTable( Q );

    [ [ 1, 2, 3, 4, 5 ],

      [ 2, 1, 4, 5, 3 ],

      [ 3, 4, 5, 1, 2 ],

      [ 4, 5, 2, 3, 1 ],

      [ 5, 3, 1, 2, 4 ] ]

    gap> elms := Elements( Q );

    gap> [ l1, l2, l3, l4, l5 ];

    gap> [ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ];

    [ l5, l4, fail ]

  
  
  
  5.4 Associators and Commutators
  
  See Section 2.5 for definitions of associators and commutators.
  
  5.4-1 Associator
  
  Associator( x, y, z )  operation
  Returns:  The associator of the elements x, y, z of the same quasigroup.
  
  5.4-2 Commutator
  
  Commutator( x, y )  operation
  Returns:  The commutator of the elements x, y of the same quasigroup.
  
  
  5.5 Generators
  
  
  5.5-1 GeneratorsOfQuasigroup and GeneratorsOfLoop
  
  GeneratorsOfQuasigroup( Q )  attribute
  GeneratorsOfLoop( Q )  attribute
  Returns:  A  set of generators of a quasigroup (resp. loop) Q. (Both methods
            are synonyms of GeneratorsOfMagma.)
  
  As  usual  in  GAP,  one can refer to the ith generator of a quasigroup Q by
  Q.i.  Note that while it is often the case that  Q.i = Elements(Q)[i], it is
  not necessarily so.
  
  5.5-2 GeneratorsSmallest
  
  GeneratorsSmallest( Q )  attribute
  Returns:  A  generating  set  {q_0,  dots, q_m} of Q such that Q_0=∅, Q_m=Q,
            Q_i=⟨ q_1, dots, q_i ⟩, and q_i+1 is the least element of Q∖ Q_i.
  
  5.5-3 SmallGeneratingSet
  
  SmallGeneratingSet( Q )  attribute
  Returns:  A  small generating set {q_0, dots, q_m} of Q obtained as follows:
            q_0  is  the  least  element for which ⟨ q_0⟩ is largest possible,
            q_1$ is the least element for which ⟨ q_0,q_1 is largest possible,
            and so on.