loops/doc/chap7.txt

  
  7 Testing Properties of Quasigroups and Loops
  
  Although  loops are quasigroups, it is often the case in the literature that
  a  property  of  the  same  name  can  differ for quasigroups and loops. For
  instance, a Steiner loop is not necessarily a Steiner quasigroup.
  
  To  avoid such ambivalences, we often include the noun Loop or Quasigroup as
  part   of  the  name  of  the  property,  e.g.,  IsSteinerQuasigroup  versus
  IsSteinerLoop.
  
  On the other hand, some properties coincide for quasigroups and loops and we
  therefore  do  not  include  Loop,  Quasigroup  as  part  of the name of the
  property, e.g., IsCommutative.
  
  
  7.1 Associativity, Commutativity and Generalizations
  
  7.1-1 IsAssociative
  
  IsAssociative( Q )  property
  Returns:  true if Q is an associative quasigroup.
  
  7.1-2 IsCommutative
  
  IsCommutative( Q )  property
  Returns:  true if Q is a commutative quasigroup.
  
  7.1-3 IsPowerAssociative
  
  IsPowerAssociative( Q )  property
  Returns:  true if Q is a power associative quasigroup.
  
  A  quasigroup  Q  is  said  to  be  power  associative if every element of Q
  generates an associative quasigroup, that is, a group.
  
  7.1-4 IsDiassociative
  
  IsDiassociative( Q )  property
  Returns:  true if Q is a diassociative quasigroup.
  
  A quasigroup Q is said to be diassociative if any two elements of Q generate
  an  associative  quasigroup,  that  is,  a  group. Note that a diassociative
  quasigroup is necessarily a loop, but it need not be so declared in LOOPS.
  
  
  7.2 Inverse Propeties
  
  For  an element x of a loop Q, the left inverse of x is the element x^λ of Q
  such  that x^λ ⋅ x = 1, while the right inverse of x is the element x^ρ of Q
  such that x⋅ x^ρ = 1.
  
  
  7.2-1 HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty
  
  HasLeftInverseProperty( Q )  property
  HasRightInverseProperty( Q )  property
  HasInverseProperty( Q )  property
  Returns:  true  if  a  loop  Q  has the left inverse property, right inverse
            property, resp. inverse property.
  
  A  loop  Q  has  the left inverse property if x^λ(xy)=y for every x, y in Q.
  Dually,  Q  has the right inverse property if (yx)x^ρ=y for every x, y in Q.
  If  Q  has both the left inverse property and the right inverse property, it
  has the inverse property.
  
  7.2-2 HasTwosidedInverses
  
  HasTwosidedInverses( Q )  property
  Returns:  true if a loop Q has two-sided inverses.
  
  A loop Q is said to have two-sided inverses if x^λ=x^ρ for every x in Q.
  
  7.2-3 HasWeakInverseProperty
  
  HasWeakInverseProperty( Q )  property
  Returns:  true if a loop Q has the weak inverse property.
  
  A  loop  Q  has  the  weak inverse property if (xy)^λ x = y^λ (equivalently,
  x(yx)^ρ = y^ρ) holds for every x, y in Q.
  
  7.2-4 HasAutomorphicInverseProperty
  
  HasAutomorphicInverseProperty( Q )  property
  Returns:  true if a loop Q has the automorphic inverse property.
  
  According  to  [Art59],  a  loop  Q  has the automorphic inverse property if
  (xy)^λ = x^λ y^λ, or, equivalently, (xy)^ρ = x^ρ y^ρ holds for every x, y in
  Q.
  
  7.2-5 HasAntiautomorphicInverseProperty
  
  HasAntiautomorphicInverseProperty( Q )  property
  Returns:  true if a loop Q has the antiautomorphic inverse property.
  
  A  loop  Q  has  the antiautomorphic inverse property if (xy)^λ=y^λ x^λ, or,
  equivalently, (xy)^ρ = y^ρ x^ρ holds for every x, y in Q.
  
  See  Appendix  B for implications implemented in LOOPS among various inverse
  properties.
  
  
  7.3 Some Properties of Quasigroups
  
  7.3-1 IsSemisymmetric
  
  IsSemisymmetric( Q )  property
  Returns:  true if Q is a semisymmetric quasigroup.
  
  A  quasigroup  Q is semisymmetric if (xy)x=y, or, equivalently x(yx)=y holds
  for every x, y in Q.
  
  7.3-2 IsTotallySymmetric
  
  IsTotallySymmetric( Q )  property
  Returns:  true if Q is a totally symmetric quasigroup.
  
  A  commutative semisymmetric quasigroup is called totally symmetric. Totally
  symmetric quasigroups are precisely the quasigroups satisfying xy=xbackslash
  y = x/y.
  
  7.3-3 IsIdempotent
  
  IsIdempotent( Q )  property
  Returns:  true if Q is an idempotent quasigroup.
  
  A quasigroup is idempotent if it satisfies x^2=x.
  
  7.3-4 IsSteinerQuasigroup
  
  IsSteinerQuasigroup( Q )  property
  Returns:  true if Q is a Steiner quasigroup.
  
  A totally symmetric idempotent quasigroup is called a Steiner quasigroup.
  
  7.3-5 IsUnipotent
  
  A quasigroup Q is unipotent if it satisfies x^2=y^2 for every x, y in Q.
  
  IsUnipotent( Q )  property
  Returns:  true if Q is a unipotent quasigroup.
  
  
  7.3-6 IsLeftDistributive, IsRightDistributive, IsDistributive
  
  IsLeftDistributive( Q )  property
  IsRightDistributive( Q )  property
  IsDistributive( Q )  property
  Returns:  true  if  Q  is  a  left  distributive  quasigroup,  resp. a right
            distributive quasigroup, resp. a distributive quasigroup.
  
  A  quasigroup  is  left distributive if it satisfies x(yz) = (xy)(xz), right
  distributive  if  it  satisfies  (xy)z = (xz)(yz), and distributive if it is
  both left distributive and right distributive.
  
  Remark: In order to be compatible with GAPs terminology, we also support the
  synonyms  IsLDistributive  and  IsRDistributive  of  IsLeftDistributive  and
  IsRightDistributive, respectively.
  
  
  7.3-7 IsEntropic and IsMedial
  
  IsEntropic( Q )  property
  IsMedial( Q )  property
  Returns:  true if Q is an entropic (aka medial) quasigroup.
  
  A  quasigroup  is entropic or medial if it satisfies the identity (xy)(uv) =
  (xu)(yv).
  
  
  7.4 Loops of Bol Moufang Type
  
  Following  [Fen69]  and  [PV05],  a  variety  of  loops  is  said  to  be of
  Bol-Moufang  type if it is defined by a single identity of Bol-Moufang type,
  i.e.,  by  an  identity  that  contains  the same 3 variables on both sides,
  exactly  one  of the variables occurs twice on both sides, and the variables
  occur in the same order on both sides.
  
  It  is  proved in [PV05] that there are 13 varieties of nonassociative loops
  of Bol-Moufang type. These are:
  
      left alternative loops defined by x(xy) = (xx)y,
  
      right alternative loops defined by x(yy) = (xy)y,
  
      left nuclear square loops defined by (xx)(yz) = ((xx)y)z,
  
      middle nuclear square loopsdefined by x((yy)z) = (x(yy))z,
  
      right nuclear square loops defined by x(y(zz)) = (xy)(zz),
  
      flexible loops defined by x(yx) = (xy)x,
  
      left   Bol   loops   defined  by  x(y(xz))  =  (x(yx))z,  always  left
        alternative,
  
      right   Bol  loops  defined  by  x((yz)y)  =  ((xy)z)y,  always  right
        alternative,
  
      LC loops defined by (xx)(yz) = (x(xy))z, always left alternative, left
        nuclear square and middle nuclear square,
  
      RC  loops  defined  by  x((yz)z) = (xy)(zz), always right alternative,
        right nuclear square and middle nuclear square,
  
      Moufang  loops  defined  by (xy)(zx) = (x(yz))x, always flexible, left
        Bol and right Bol,
  
      C loops defined by x(y(yz)) = ((xy)y)z, always LC and RC,
  
      extra loops defined by x(y(zx)) = ((xy)z)x, always Moufang and C.
  
  Note  that  although  some of the defining identities are not of Bol-Moufang
  type,  they  are  equivalent  to  a  Bol-Moufang  identity.  Moreover,  many
  varieties  of  loops  of  Bol-Moufang  type can be defined by one of several
  equivalent identities of Bol-Moufang type.
  
  There  are  also  several  varieties related to loops of Bol-Moufang type. A
  loop  is  said  to  be  alternative if it is both left alternative and right
  alternative.  A  loop is nuclear square if it is left nuclear square, middle
  nuclear square and right nuclear square.
  
  7.4-1 IsExtraLoop
  
  IsExtraLoop( Q )  property
  Returns:  true if Q is an extra loop.
  
  7.4-2 IsMoufangLoop
  
  IsMoufangLoop( Q )  property
  Returns:  true if Q is a Moufang loop.
  
  7.4-3 IsCLoop
  
  IsCLoop( Q )  property
  Returns:  true if Q is a C loop.
  
  7.4-4 IsLeftBolLoop
  
  IsLeftBolLoop( Q )  property
  Returns:  true if Q is a left Bol loop.
  
  7.4-5 IsRightBolLoop
  
  IsRightBolLoop( Q )  property
  Returns:  true if Q is a right Bol loop.
  
  7.4-6 IsLCLoop
  
  IsLCLoop( Q )  property
  Returns:  true if Q is an LC loop.
  
  7.4-7 IsRCLoop
  
  IsRCLoop( Q )  property
  Returns:  true if Q is an RC loop.
  
  7.4-8 IsLeftNuclearSquareLoop
  
  IsLeftNuclearSquareLoop( Q )  property
  Returns:  true if Q is a left nuclear square loop.
  
  7.4-9 IsMiddleNuclearSquareLoop
  
  IsMiddleNuclearSquareLoop( Q )  property
  Returns:  true if Q is a middle nuclear square loop.
  
  7.4-10 IsRightNuclearSquareLoop
  
  IsRightNuclearSquareLoop( Q )  property
  Returns:  true if Q is a right nuclear square loop.
  
  7.4-11 IsNuclearSquareLoop
  
  IsNuclearSquareLoop( Q )  property
  Returns:  true if Q is a nuclear square loop.
  
  7.4-12 IsFlexible
  
  IsFlexible( Q )  property
  Returns:  true if Q is a flexible quasigroup.
  
  7.4-13 IsLeftAlternative
  
  IsLeftAlternative( Q )  property
  Returns:  true if Q is a left alternative quasigroup.
  
  7.4-14 IsRightAlternative
  
  IsRightAlternative( Q )  property
  Returns:  true if Q is a right alternative quasigroup.
  
  7.4-15 IsAlternative
  
  IsAlternative( Q )  property
  Returns:  true if Q is an alternative quasigroup.
  
  While  listing  the  varieties  of  loops  of Bol-Moufang type, we have also
  listed  all  inclusions among them. These inclusions are built into LOOPS as
  filters.
  
  The  following trivial example shows some of the implications and the naming
  conventions of LOOPS at work:
  
    Example  
    gap> L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] );

    <loop of order 2>

    gap> [ IsLeftBolLoop( L ), L ]

    [ true, <left Bol loop of order 2> ]

    gap> [ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ];

    [ true, true ]

    gap> [ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ];

    [ false, true ]

    gap> L;

    <Moufang loop of order 2>

    gap> [ IsAssociative( L ), L ];

    [ true, <associative loop of order 2> ]

  
  
  The  analogous  terminology  for  quasigroups  of  Bol-Moufang  type  is not
  standard  yet, and hence is not supported in LOOPS except for the situations
  explicitly noted above.
  
  
  7.5 Power Alternative Loops
  
  A  loop  is  left power alternative if it is power associative and satisfies
  x^n(x^m  y) = x^n+my for all elements x, y and all integers m, n. Similarly,
  a  loop  is right power alternative if it is power associative and satisfies
  (x  y^n)y^m  = xy^n+m for all elements x, y and all integers m, n. A loop is
  power  alternative  if  it  is  both  left power alternative and right power
  alternative.
  
  Left  power alternative loops are left alternative and have the left inverse
  property. Left Bol loops and LC loops are left power alternative.
  
  
  7.5-1 IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative
  
  IsLeftPowerAlternative( Q )  property
  IsRightPowerAlternative( Q )  property
  IsPowerAlternative( Q )  property
  Returns:  true  if  Q  is a left power alternative loop, resp. a right power
            alternative loop, resp. a power alternative loop.
  
  
  7.6 Conjugacy Closed Loops and Related Properties
  
  A  loop  Q  is left conjugacy closed if the set of left translations of Q is
  closed under conjugation (by itself). Similarly, a loop Q is right conjugacy
  closed  if the set of right translations of Q is closed under conjugation. A
  loop  is  conjugacy  closed  if  it  is both left conjugacy closed and right
  conjugacy  closed.  It is common to refer to these loops as LCC, RCC, and CC
  loops, respectively.
  
  The equivalence LCC + RCC = CC is built into LOOPS.
  
  7.6-1 IsLCCLoop
  
  IsLCCLoop( Q )  property
  IsLeftConjugacyClosedLoop( Q )  property
  Returns:  true if Q is a left conjugacy closed loop.
  
  7.6-2 IsRCCLoop
  
  IsRCCLoop( Q )  property
  IsRightConjugacyClosedLoop( Q )  property
  Returns:  true if Q is a right conjugacy closed loop.
  
  7.6-3 IsCCLoop
  
  IsCCLoop( Q )  property
  IsConjugacyClosedLoop( Q )  property
  Returns:  true if Q is a conjugacy closed loop.
  
  7.6-4 IsOsbornLoop
  
  IsOsbornLoop( Q )  property
  Returns:  true if Q is an Osborn loop.
  
  A loop is Osborn if it satisfies x(yz⋅ x)=(x^λbackslash y)(zx). Both Moufang
  loops and CC loops are Osborn.
  
  
  7.7 Automorphic Loops
  
  A  loop  Q  whose  all  left  (resp. middle, resp. right) inner mappings are
  automorphisms  of  Q  is  known  as  a  left  automorphic loop (resp. middle
  automorphic loop, resp. right automorphic loop).
  
  A loop Q is an automorphic loop if all inner mappings of Q are automorphisms
  of Q.
  
  Automorphic  loops are also known as A loops, and similar terminology exists
  for left, right and middle automorphic loops.
  
  The following results hold for automorphic loops:
  
      automorphic loops are power associative [BP56]
  
      in  an  automorphic  loop  Q  we  have  Nuc(Q) = Nuc_λ(Q) = Nuc_ρ(Q)le
        Nuc_μ(Q) and all nuclei are normal [BP56]
  
      a  loop  that  is left automorphic and right automorphic satisfies the
        anti-automorphic inverse property and is automorphic [JKNV11]
  
      diassociative automorphic loops are Moufang [KKP02]
  
      automorphic loops of odd order are solvable [KKPV16]
  
      finite commutative automorphic loops are solvable [GKN14]
  
      commutative  automorphic  loops of order p, 2p, 4p, p^2, 2p^2, 4p^2 (p
        an odd prime) are abelian groups [Voj15]
  
      commutative  automorphic  loops of odd prime power order are centrally
        nilpotent [JKV12]
  
      for  any  prime  p, there are 7 commutative automorphic loops of order
        p^3 up to isomorphism [BGV12]
  
  See the built-in filters and the survey [Voj15] for additional properties of
  automorphic loops.
  
  7.7-1 IsLeftAutomorphicLoop
  
  IsLeftAutomorphicLoop( Q )  property
  IsLeftALoop( Q )  property
  Returns:  true if Q is a left automorphic loop.
  
  7.7-2 IsMiddleAutomorphicLoop
  
  IsMiddleAutomorphicLoop( Q )  property
  IsMiddleALoop( Q )  property
  Returns:  true if Q is a middle automorphic loop.
  
  7.7-3 IsRightAutomorphicLoop
  
  IsRightAutomorphicLoop( Q )  property
  IsRightALoop( Q )  property
  Returns:  true if Q is a right automorphic loop.
  
  7.7-4 IsAutomorphicLoop
  
  IsAutomorphicLoop( Q )  property
  IsALoop( Q )  property
  Returns:  true if Q is an automorphic loop.
  
  Remark: Be careful not to confuse IsALoop and IsLoop.
  
  
  7.8 Additonal Varieties of Loops
  
  7.8-1 IsCodeLoop
  
  IsCodeLoop( Q )  property
  Returns:  true if Q is a code loop.
  
  A  code  loop  is  a Moufang 2-loop with a Frattini subloop of order 1 or 2.
  Code loops are extra and conjugacy closed.
  
  7.8-2 IsSteinerLoop
  
  IsSteinerLoop( Q )  property
  Returns:  true if Q is a Steiner loop.
  
  A  Steiner loop is an inverse property loop of exponent 2. Steiner loops are
  commutative.
  
  
  7.8-3 IsLeftBruckLoop and IsLeftKLoop
  
  IsLeftBruckLoop( Q )  property
  IsLeftKLoop( Q )  property
  Returns:  true if Q is a left Bruck loop (aka left K loop).
  
  A  left  Bol  loop  with the automorphic inverse property is known as a left
  Bruck loop or a left K loop.
  
  
  7.8-4 IsRightBruckLoop and IsRightKLoop
  
  IsRightBruckLoop( Q )  property
  IsRightKLoop( Q )  property
  Returns:  true if Q is a right Bruck loop (aka right K loop).
  
  A  right  Bol loop with the automorphic inverse property is known as a right
  Bruck loop or a right K loop.