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RightQuasigroups

Computing with one-sided quasigroups in GAP.

0.9.1

28 June 2023

Gábor P. Nagy
Email: nagyg@math.u-szeged.hu
Homepage: http://www.math.u-szeged.hu/~nagyg
Address:
Bolyai Institute of the University of Szeged
Aradi vértanúk tere 1
H-6720 Szeged (Hungary)

Petr Vojtěchovský
Email: petr@math.du.edu
Homepage: http://www.math.du.edu/~petr/
Address:
Department of Mathematics, University of Denver
2390 S York St
Denver, CO 80208
USA

Contents

1.1 Main features of the package

1.2 Categories and representations for right quasigroups, quasigroups and loops

  1.2-1 Categories and representations for right quasigroup elements and right quasigroups
  1.2-2 Categories and representations for quasigroup elements and quasigroups
  1.2-3 Categories and representations for loop elements and loops
  1.2-4 CategoryOfRightQuasigroup

1.3 Displaying right quasigroups and their elements

  1.3-1 Displaying right quasigroups, quasigroups and loops
  1.3-2 Displaying right quasigroup elements
  1.3-3 Changing the name of right quasigroup elements

1.4 The underlying set

  1.4-1 UnderlyingSetElm
  1.4-2 UnderlyingSet
  1.4-3 ChangeUnderlyingSet

1.5 Accessing elements and elementwise right quasigroup operations

  1.5-1 RightQuotient
  1.5-2 LeftQuotient
  1.5-3 Inverses in loops
  1.5-4 Powers, order and exponent
  1.5-5 Commutator
  1.5-6 Associator

1.6 First examples

1.7 The parent right quasigroup

1.8 Index based and canonical right quasigroups

  1.8-1 ParentInd
  1.8-2 IsIndexBased
  1.8-3 IndexBasedCopy
  1.8-4 IsCanonical
  1.8-5 CanonicalCopy

1.9 Generators and comparison of right quasigroups

1.9-1 SmallGeneratingSet
1.9-2 GeneratorsSmallest

1.10 Non-qualified operations

1.11 *Direct access to the parent right quasigroup record

2.1 Optional arguments in constructors

2.1-1 SetDefaultConstructorStyle
2.1-2 ConstructorStyle

2.2 Converting betwen right quasigroups, quasigroups and loops

2.2-1 IsRightQuasigroupMagma, IsLeftQuasigroupMagma, IsQuasigroupMagma and IsLoopMagma
2.2-2 AsRightQuasigroup, AsQuasigroup and AsLoop

2.3 *A two-step constructor

2.3-1 RQ_AlgebraShell
2.3-2 RQ_AddDefaultOperations

2.4 Constructors by Cayley table

  2.4-1 Cayley tables versus multiplication tables
  2.4-2 Cayley tables and multiplication tables of right quasigroups
  2.4-3 Constructing division tables from multiplication tables
  2.4-4 Testing Cayley tables
  2.4-5 Normalizing Cayley tables
  2.4-6 Creating right quasigroups by Cayley table

2.5 Constructors by function

  2.5-1 Arguments used in constructors by function
  2.5-2 IsRightQuasigroupFunction
  2.5-3 IsQuasigroupFunction
  2.5-4 IsLoopFunction
  2.5-5 Multiplication, right division and left division functions
  2.5-6 RightQuasigroupByFunction
  2.5-7 QuasigroupByFunction
  2.5-8 LoopByFunction

2.6 Constructors by right section

2.6-1 Testing right sections
2.6-2 Constructing right quasigroups, quasigroups and loops by right section

2.7 Constructors by right folder (transversal)

2.7-1 Checking right folders
2.7-2 Creating right quasigroups, quasigroups and loops by right folders

2.8 Constructors for special types of right quasigroups

2.8-1 ProjectionRightQuasigroup

2.9 Random right quasigroups, quasigroups and loops

  2.9-1 RandomRightQuasigroup
  2.9-2 Random quasigroups and loops
  2.9-3 RandomNilpotentLoop

3 Direct products, subalgebras and factor algebras

3.1 Direct product of right quasigroups

3.2 Opposite quasigroups

3.2-1 Opposite quasigroups and loops

3.3 Subalgebras

  3.3-1 Testing for subalgebras
  3.3-2 Constructing subalgebras
  3.3-3 All subalgebras
  3.3-4 Testing minimal subalgebras
  3.3-5 All minimal subalgebras
  3.3-6 Testing maximal subalgebras
  3.3-7 All maximal subalgebras

3.4 Cosets and transversals

3.4-1 RightTransversal
3.4-2 LeftTransversal

3.5 Right quasigroups by generators

  3.5-1 RightQuasigroup, Quasigroup and Loop
  3.5-2 RightQuasigroupByGenerators, QuasigroupByGenerators and LoopByGenerators
  3.5-3 RightQuasigroupWithGenerators, QuasigroupWithGenerators and LoopWithGenerators

3.6 Intersections and joins of right quasigroups

3.7 Congruences

  3.7-1 Checking right quasigroup congruences
  3.7-2 Congruences generated by partition
  3.7-3 Congruences generated by pairs
  3.7-4 All congruences

3.8 Normality and simplicity

3.8-1 AllNormalSubloops
3.8-2 Testing right quasigroups for simplicity

3.9 Factor algebras

3.9-1 Constructing factor algebras

3.10 An example of the factor construction: Paige loops

4 Mappings, transformations and permutations of right quasigroups

4.1 Mappings, transformations and permutations

4.2 Right quasigroup mapppings

4.2-1 IsRightQuasigroupMapping, IsQuasigroupMapping, IsLoopMapping
4.2-2 AsRightQuasigroupMapping

4.3 Canonical permutations and parent permutations

  4.3-1 IsCanonicalPerm
  4.3-2 IsParentPerm
  4.3-3 AsCanonicalPerm
  4.3-4 AsParentPerm

4.4 Canonical transformations and parent transformations

  4.4-1 IsCanonicalTransformation
  4.4-2 IsParentTransformation
  4.4-3 AsCanonicalTransformation
  4.4-4 AsParentTransformation

5 Associated permutation groups

5.1 Translations and sections

  5.1-1 RightTranslation
  5.1-2 LeftTranslation
  5.1-3 Sections

5.2 Multiplication groups and relative multiplication groups

  5.2-1 Multiplication groups
  5.2-2 LeftMultiplicationSemigroup
  5.2-3 Relative multiplication groups

5.3 Inner mappings and inner mapping groups

5.3-1 Inner mappings
5.3-2 Inner mapping groups

5.4 Displacement groups

  5.4-1 Right displacement groups
  5.4-2 Left displacement groups
  5.4-3 IsIsotopicToGroup
  5.4-4 IsIsotopicToAbelianGroup

5.5 Realizing permutation groups as multiplication groups of loops

  5.5-1 AllLoopsWithMltInGroup
  5.5-2 AllNonassociativeLoopsWithMltInGroup
  5.5-3 OneLoopWithMltInGroup
  5.5-4 OneNonassociativeLoopWithMltInGroup
  5.5-5 AllLoopsWithMltGroup
  5.5-6 OneLoopWithMltGroup

6 Nilpotency and solvability

6.1 Nuclei, commutant and center

  6.1-1 Nuclei
  6.1-2 Commutant
  6.1-3 Center

6.2 Nilpotency for loops

  6.2-1 NilpotencyClassOfLoop
  6.2-2 IsNilpotentLoop
  6.2-3 UpperCentralSeriesOfLoop
  6.2-4 LowerCentralSeriesOfLoop

6.3 Solvability and congruence solvability for loops

  6.3-1 AssociatorSubloop
  6.3-2 IsSolvableLoop
  6.3-3 DerivedSubloop
  6.3-4 DerivedSeriesOfLoop
  6.3-5 DerivedLength
  6.3-6 CommutatorOfNormalSubloops
  6.3-7 IsAbelianNormalSubloop
  6.3-8 IsCongruenceSolvableLoop
  6.3-9 CongruenceDerivedSeriesOfLoop
  6.3-10 CongruenceDerivedLength

6.4 Nuclear and central extensions of loops

  6.4-1 IsLoopCocycle
  6.4-2 LoopByNuclearExtension
  6.4-3 LoopByCentralExtension
  6.4-4 AsSquareTable
  6.4-5 NuclearExtensionByNormalSubloop
  6.4-6 CentralExtensionByNormalSubloop

6.5 All central extensions in a variety of loops

  6.5-1 LoopCocyclesInVariety
  6.5-2 LoopCoboundaries
  6.5-3 LoopCocyclesModAction
  6.5-4 AllLoopCocyclesInVariety
  6.5-5 AllLoopCentralExtensionsInVariety

6.6 Frattini subalgebra

  6.6-1 FrattiniSubrightquasigroup
  6.6-2 FrattiniSubquasigroup
  6.6-3 FrattiniSubloop

7 Properties of right quasigroups, quasigroups and loops

  7.1-1 Building terms from strings
  7.1-2 Verifying identities
  7.1-3 *How evaluation of identities with the parser works

7.2 Properties of magmas

7.2-1 Testing properties of magmas and right quasigroups

7.3 Properties of right quasigroups

7.3-1 IsFaithfulRightQuasigroup
7.3-2 IsProjectionRightQuasigroup

7.4 Properties of quasigroups

  7.4-1 IsSemisymmetric
  7.4-2 IsTotallySymmetric
  7.4-3 IsSteinerQuasigroup
  7.4-4 IsEntropic

7.5 Inverse properties of loops

  7.5-1 HasTwosidedInverses
  7.5-2 Right and left inverse properties
  7.5-3 HasWeakInverseProperty
  7.5-4 Automorphic and antiautomorphic inverse properties

7.6 Loops of Bol-Moufang type

7.6-1 Testing for loops of Bol-Moufang type
7.6-2 IsNuclearSquareLoop

7.7 Power alternative loops

7.7-1 Testing for power alternative loops

7.8 Conjugacy closed loops and Osborn loops

7.8-1 Testing for conjugacy closed loops
7.8-2 IsOsbornLoop

7.9 Additional varieties of loops

  7.9-1 IsCodeLoop
  7.9-2 IsSteinerLoop
  7.9-3 Bruck loops
  7.9-4 Automorphic loops

8 Homomorphisms, isomorphisms and automorphisms

8.1 Homomorphisms, isomorphisms and automorphisms of right quasigroups

  8.1-1 Testing homomorphisms
  8.1-2 Testing isomorphisms
  8.1-3 Testing endomorphisms
  8.1-4 Testing automorphisms
  8.1-5 Homomorphisms by images
  8.1-6 KernelRelationOfMapping
  8.1-7 Kernel of loop homomorphisms
  8.1-8 Natural homomorphism by congruence
  8.1-9 Natural homomorphism by normal subloop

8.2 Isomorphs of right quasigroups

8.2-1 Isomorphs

8.3 Right quasigroups up to isomorphisms

  8.3-1 IsomorphismDiscriminator
  8.3-2 AreEqualIsomorphismDiscriminators
  8.3-3 IsomorphismRightQuasigroups, IsomorphismQuasigroups, IsomorphismLoops
  8.3-4 Right quasigroups up to isomorphism

8.4 Automorphism groups of right quasigroups

8.4-1 AutomorphismGroup

9 Homotopisms, isotopisms and autotopisms

9.1 Homotopisms, isotopism and autotopisms of right quasigroups

  9.1-1 Representing homotopisms in GAP
  9.1-2 Attributes of homotopisms
  9.1-3 Properties of homotopisms
  9.1-4 Creating homotopisms
  9.1-5 Creating autotopisms
  9.1-6 Operations with homotopisms
  9.1-7 Testing homotopism data
  9.1-8 Testing isotopism data
  9.1-9 Testing autotopism data

9.2 Twists of right quasigroups

9.2-1 RightQuasigroupTwist, QuasigroupTwist and LoopTwist

9.3 Isotopes of right quasigroups

  9.3-1 RightQuasigroupIsotope
  9.3-2 QuasigroupIsotope
  9.3-3 LoopIsotope
  9.3-4 PrincipalLoopIsotope

9.4 Affine right quasigroups

  9.4-1 IsAffineRightQuasigroupArithmeticForm
  9.4-2 IsAffineQuasigroupArithmeticForm
  9.4-3 AffineRightQuasigroup
  9.4-4 AffineQuasigroup

9.5 Right quasigroups up to isotopism

  9.5-1 IsotopismDiscriminator
  9.5-2 AreEqualIsotopismDiscriminators
  9.5-3 ArePossiblyIsotopicRightQuasigroups
  9.5-4 IsotopismRightQuasigroups
  9.5-5 IsotopismQuasigroups
  9.5-6 IsotopismLoops
  9.5-7 Right quasigroups up to isotopism

9.6 Autotopism groups of right quasigroups

  9.6-1 RQ_NiceMonomorphism
  9.6-2 RQ_NiceMonomorphismInverse
  9.6-3 RQ_AutotopismGroupByGeneratorsNC
  9.6-4 AutotopismGroup

9.7 Topisms using the associated colored digraph

9.7-1 RQ_Digraph
9.7-2 RQ_BlissCanonicalLabeling4Morphism

10 Bol loops, Bruck loops and Moufang loops

10.1 Bol loops and Bruck loops

  10.1-1 Associated Bruck loop
  10.1-2 IsExactGroupFactorization
  10.1-3 RightBolLoopByExactGroupFactorization

10.2 Triality for Moufang loops

10.2-1 TrialityPermGroup
10.2-2 TrialityPcGroup

10.3 Quarter modifications of Moufang loops

  10.3-1 CheinLoop
  10.3-2 *Cyclic modification
  10.3-3 *Dihedral modification

11 Racks and quandles

11.1 Testing properties of racks and quandles

  11.1-1 IsRack and IsQuandle
  11.1-2 IsHomogeneousRack and IsHomogeneousQuandle
  11.1-3 IsConnectedRack and IsConnectedQuandle
  11.1-4 IsLatinRack and IsLatinQuandle
  11.1-5 IsProjectionRack and IsProjectionQuandle
  11.1-6 IsPermutationalRack and IsPermutationalQuandle
  11.1-7 IsFaithfulRack and IsFaithfulQuandle

11.2 Constructors for racks

  11.2-1 PermutationalRack
  11.2-2 CyclicRack
  11.2-3 IsAffineRackArithmeticForm
  11.2-4 AffineRack

11.3 Constructors for quandles

  11.3-1 IsAffineQuandleArithmeticForm
  11.3-2 AffineQuandle
  11.3-3 DihedralQuandle
  11.3-4 CoreOfGroup
  11.3-5 CoreOfRightBolLoop
  11.3-6 GalkinQuandle and HomogeneousQuandle
  11.3-7 ConjugationQuandle

11.4 Rack envelopes and quandle envelopes: The Joyce-Blackburn representation

  11.4-1 IsRackEnvelope and IsQuandleEnvelope
  11.4-2 RackEnvelope and QuandleEnvelope
  11.4-3 RackByRackEnvelope

11.5 Subracks and subquandles

  11.5-1 Testing subracks and subquandles
  11.5-2 Creating subracks and subquandles
  11.5-3 All subracks and subquandles
  11.5-4 Racks and quandles by generators

11.6 Isomorphisms and isotopisms of racks and quandles

11.6-1 IsomorphismRacks, IsomorphismQuandles
11.6-2 Racks and quandles up to isomorphism

11.7 Groups associated with racks and quandles

11.7-1 AdjointGroup

12 Libraries of loops, racks and quandles

12.1 *A typical library

12.2 Accessing algebras in the library

  12.2-1 DisplayLibraryInfo
  12.2-2 LibraryAlgebra
  12.2-3 LibraryAlgebras

12.3 Library of left Bol loops and right Bol loops

12.3-1 LeftBolLoop and RightBolLoop
12.3-2 LeftBolLoops and RightBolLoops

12.4 Library of left Bruck loops and right Bruck loops

12.4-1 LeftBruckLoop and RightBruckLoop
12.4-2 LeftBruckLoops and RightBruckLoops

12.5 Library of Moufang Loops

12.5-1 MoufangLoop
12.5-2 MoufangLoops

12.6 Library of code loops

12.6-1 CodeLoop
12.6-2 CodeLoops

12.7 Library of Steiner loops

12.7-1 SteinerLoop
12.7-2 SteinerLoops

12.8 Library of conjugacy closed loops

  12.8-1 LCCLoop, LeftConjugacyClosedLoop, RCCLoop and RightConjugacyClosedLoop
  12.8-2 LCCLoop, LeftConjugacyClosedLoop, RCCLoop and RightConjugacyClosedLoop
  12.8-3 CCLoop and ConjugacyClosedLoop
  12.8-4 CCLoops and ConjugacyClosedLoops

12.9 Library of small loops

12.9-1 SmallLoop
12.9-2 SmallLoops

12.10 Library of Paige loops

12.10-1 PaigeLoop
12.10-2 PaigeLoops

12.11 Library of nilpotent loops

12.11-1 NilpotentLoop
12.11-2 NilpotentLoops

12.12 Library of automorphic loops

12.12-1 AutomorphicLoop
12.12-2 AutomorphicLoops

12.13 Library of interesting loops

12.13-1 InterestingLoop
12.13-2 InterestingLoops

12.14 Library of small racks

12.14-1 SmallRack
12.14-2 SmallRacks

12.15 Library of small quandles

12.15-1 SmallQuandle
12.15-2 SmallQuandles

12.16 Library of connected quandles

12.16-1 ConnectedQuandle
12.16-2 ConnectedQuandles

12.17 Library of small loops up to isotopism

12.17-1 ItpSmallLoop
12.17-2 ItpSmallLoops

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