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
.
‣ IsAssociative ( Q ) | ( property ) |
Returns: true
if Q is an associative quasigroup.
‣ IsCommutative ( Q ) | ( property ) |
Returns: true
if Q is a commutative quasigroup.
‣ 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.
‣ 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.
For an element \(x\) of a loop \(Q\), the left inverse of \(x\) is the element \(x^\lambda\) of \(Q\) such that \(x^\lambda \cdot x = 1\), while the right inverse of \(x\) is the element \(x^\rho\) of \(Q\) such that \(x\cdot x^\rho = 1\).
‣ 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 \(x^\lambda(xy)=y\) for every \(x\), \(y\) in \(Q\). Dually, \(Q\) has the right inverse property if \((yx)x^\rho=y\) for every \(x\), \(y\) in \(Q\). If \(Q\) has both the left inverse property and the right inverse property, it has the inverse property.
‣ HasTwosidedInverses ( Q ) | ( property ) |
Returns: true
if a loop Q has two-sided inverses.
A loop \(Q\) is said to have two-sided inverses if \(x^\lambda=x^\rho\) for every \(x\) in \(Q\).
‣ HasWeakInverseProperty ( Q ) | ( property ) |
Returns: true
if a loop Q has the weak inverse property.
A loop \(Q\) has the weak inverse property if \((xy)^\lambda x = y^\lambda\) (equivalently, \(x(yx)^\rho = y^\rho\)) holds for every \(x\), \(y\) in \(Q\).
‣ 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 \((xy)^\lambda = x^\lambda y^\lambda\), or, equivalently, \((xy)^\rho = x^\rho y^\rho\) holds for every \(x\), \(y\) in \(Q\).
‣ HasAntiautomorphicInverseProperty ( Q ) | ( property ) |
Returns: true
if a loop Q has the antiautomorphic inverse property.
A loop \(Q\) has the antiautomorphic inverse property if \((xy)^\lambda=y^\lambda x^\lambda\), or, equivalently, \((xy)^\rho = y^\rho x^\rho\) holds for every \(x\), \(y\) in \(Q\).
See Appendix B for implications implemented in LOOPS among various inverse properties.
‣ 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 \(Q\).
‣ 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 \(xy=x\backslash y = x/y\).
‣ IsIdempotent ( Q ) | ( property ) |
Returns: true
if Q is an idempotent quasigroup.
A quasigroup is idempotent if it satisfies \(x^2=x\).
‣ IsSteinerQuasigroup ( Q ) | ( property ) |
Returns: true
if Q is a Steiner quasigroup.
A totally symmetric idempotent quasigroup is called a Steiner quasigroup.
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.
‣ 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.
‣ 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 \((xy)(uv) = (xu)(yv)\).
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.
‣ IsExtraLoop ( Q ) | ( property ) |
Returns: true
if Q is an extra loop.
‣ IsMoufangLoop ( Q ) | ( property ) |
Returns: true
if Q is a Moufang loop.
‣ IsCLoop ( Q ) | ( property ) |
Returns: true
if Q is a C loop.
‣ IsLeftBolLoop ( Q ) | ( property ) |
Returns: true
if Q is a left Bol loop.
‣ IsRightBolLoop ( Q ) | ( property ) |
Returns: true
if Q is a right Bol loop.
‣ IsLCLoop ( Q ) | ( property ) |
Returns: true
if Q is an LC loop.
‣ IsRCLoop ( Q ) | ( property ) |
Returns: true
if Q is an RC loop.
‣ IsLeftNuclearSquareLoop ( Q ) | ( property ) |
Returns: true
if Q is a left nuclear square loop.
‣ IsMiddleNuclearSquareLoop ( Q ) | ( property ) |
Returns: true
if Q is a middle nuclear square loop.
‣ IsRightNuclearSquareLoop ( Q ) | ( property ) |
Returns: true
if Q is a right nuclear square loop.
‣ IsNuclearSquareLoop ( Q ) | ( property ) |
Returns: true
if Q is a nuclear square loop.
‣ IsFlexible ( Q ) | ( property ) |
Returns: true
if Q is a flexible quasigroup.
‣ IsLeftAlternative ( Q ) | ( property ) |
Returns: true
if Q is a left alternative quasigroup.
‣ IsRightAlternative ( Q ) | ( property ) |
Returns: true
if Q is a right alternative quasigroup.
‣ 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:
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.
A loop is left power alternative if it is power associative and satisfies \(x^n(x^m y) = x^{n+m}y\) 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.
‣ 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.
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.
‣ IsLCCLoop ( Q ) | ( property ) |
‣ IsLeftConjugacyClosedLoop ( Q ) | ( property ) |
Returns: true
if Q is a left conjugacy closed loop.
‣ IsRCCLoop ( Q ) | ( property ) |
‣ IsRightConjugacyClosedLoop ( Q ) | ( property ) |
Returns: true
if Q is a right conjugacy closed loop.
‣ IsCCLoop ( Q ) | ( property ) |
‣ IsConjugacyClosedLoop ( Q ) | ( property ) |
Returns: true
if Q is a conjugacy closed loop.
‣ IsOsbornLoop ( Q ) | ( property ) |
Returns: true
if Q is an Osborn loop.
A loop is Osborn if it satisfies \(x(yz\cdot x)=(x^\lambda\backslash y)(zx)\). Both Moufang loops and CC loops are Osborn.
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 \({\rm Nuc}(Q) = {\rm Nuc}_{\lambda}(Q) = {\rm Nuc}_{\rho}(Q)\le {\rm Nuc}_{\mu}(Q)\) 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.
‣ IsLeftAutomorphicLoop ( Q ) | ( property ) |
‣ IsLeftALoop ( Q ) | ( property ) |
Returns: true
if Q is a left automorphic loop.
‣ IsMiddleAutomorphicLoop ( Q ) | ( property ) |
‣ IsMiddleALoop ( Q ) | ( property ) |
Returns: true
if Q is a middle automorphic loop.
‣ IsRightAutomorphicLoop ( Q ) | ( property ) |
‣ IsRightALoop ( Q ) | ( property ) |
Returns: true
if Q is a right automorphic loop.
‣ IsAutomorphicLoop ( Q ) | ( property ) |
‣ IsALoop ( Q ) | ( property ) |
Returns: true
if Q is an automorphic loop.
Remark: Be careful not to confuse IsALoop
and IsLoop
.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
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