Chapter 7: Mappings in RightQuasigroups

  • Mappings vs functions
  • Parent vs canonical
  • Isomorphisms vs automorphisms

Mappings vs functions

  • A mapping in GAP is what is called a "function" in mathematics

  • The domain of is called source
  • The codomain is called range
  • The image is called image
  • The image of under the mapping is expressed as , or Image(f,x)
    gap> A:=Domain([1..100]); B:=Domain([0..15]);
Domain([ 1 .. 100 ])
Domain([ 0 .. 15 ])
gap> f:=MappingByFunction(A,B,x->(x^2) mod 10);
MappingByFunction( Domain([ 1 .. 100 ]), Domain([ 0 .. 15 ]), ... )
gap> 6^f;
6
gap> Image(f);
[ 0, 1, 4, 5, 6, 9 ]

Composition of mappings

Mappings can be composed by the operator "*"

gap> C:=GF(2);
GF(2)
gap> g:=MappingByFunction(B,C,x->Z(2)*x);
MappingByFunction( Domain([ 0 .. 15 ]), GF(2), function( x ) ... end )
gap> f*g;
CompositionMapping( MappingByFunction( Domain(
[ 0 .. 15 ]), GF(2), function( x ) ... end ), MappingByFunction( Domain(
[ 1 .. 100 ]), Domain([ 0 .. 15 ]), function( x ) ... end ) )
gap> g*f;
Error, no method found! ...

gap> Source(f*g); Range(f*g); Image(f*g);
Domain([ 1 .. 100 ])
GF(2)
[ 0*Z(2), Z(2)^0 ]
gap> IsSurjective(f*g) and not IsInjective(f*g);
true

Parent and canonical permutations

For a fixed right quasigroup Q, permutations can be understood in two ways:

Parent permutations

  • Indexing is based on the parent indices of elements of Q.
  • More precisely, a permutation f is a parent permutation of Q if it restricts to a permutation of the set ParentInd( Q ).
    gap> AsParentPerm(T,(3,6));
(6,15)

Canonical permutations

  • Indexing is based on the position of elements in Elements( Q ).
  • More precisely, a permutation f is a canonical permutation on Q if it restricts to a permutation on [1..Size(Q)].
    gap> AsCanonicalPerm(T,(3,6));
(2,3)

A permutation does not keep track of Q, the right quasigroup Q must be provided.

Parent and canonical transformations

For fixed right quasigroups Q1, Q2, transformations can be understood in two ways:

Parent transformations

  • Indexing is based on the parent indices of elements of Q1 and Q2.
  • More precisely, a transformation t is a parent transformation from Q1 to Q2 if for every i in ParentInt( Q1 ) we have i^t in ParentInd( Q2 ).

Canonical transformations

  • Indexing is based on the position of elements among elements of Q1 and Q2.
  • More precisely, a transformation t is a canonical transformation from Q1 to Q2 if for every i in [1..Size(Q1)] we have i^t in [1..Size(Q2)].

A transformation does not keep track of Q1 and Q2. The right quasigroups Q1 and Q2 must therefore be provided.

Right quasigroup mappings

Given right quasigroups Q1 and Q2, a mapping from Q1 to Q2 is represented in one of the following ways:

  • as a GAP mapping with source Q1 and range Q2 (used for mappings between two distinct right quasigroups, for instance for homomorphisms),
  • as a transformation (for instance for left translations of right quasigroups),
    gap> Q:=AffineQuandle(10,3);
<quandle of size 10>
gap> LeftTranslation(Q,Q.3);
Transformation( [ 7, 5, 3, 1, 9, 7, 5, 3, 1, 9 ] )
  • as a permutation (used for bijective mappings when Q1 = Q2, for instance for translations in quasigroups, automorphisms, etc.).
    gap> RightTranslation(Q,Q.3);
(1,7,5,9)(2,10,4,6)

Example: Chein loop

  • Let be a group. The Chein loop is a Moufang loop on .
  • Generic elements are labeled by , where and .
    gap> G:=SmallGroup(21,1); StructureDescription(G);
<pc group of size 21 with 2 generators>
"C7 : C3"
gap> M:=CheinLoop(G);
<Moufang loop of size 42>
  • We construct the embedding 
    gap> emb:=MappingByFunction(G,M,x->M[[0,x]]);
MappingByFunction( C7 : C3, <Moufang loop of size 42>, \ 
function( x ) ... end )
gap> RespectsMultiplication(emb);
true

Chein loop element indexing

Remainder: R.i refers to the index element in the parent right quasigroup of R

gap> S:=Subloop(M,Image(emb,GeneratorsOfGroup(G)));
<Moufang loop of size 21>
gap> T:=DerivedSubloop(S);
<Moufang loop of size 7>
gap> ParentInd(T);
[ 1, 3, 6, 9, 12, 15, 18 ]
gap> T.3=S.3 and T.3=M.3 and not T.3=Elements(T)[3];
true

Automorphisms

  • Automorphisms of a right quasigroup are represented as parent permutations
gap> G; S; T; ParentInd(T);
C7 : C3
<Moufang loop of size 21>
<Moufang loop of size 7>
[ 1, 3, 6, 9, 12, 15, 18 ]
gap> AutomorphismGroup(T);
Group([ (3,6,12)(9,18,15), (3,9,6,18,12,15) ])

Isomorphisms

  • Isomorphisms (and more generally homomorphisms) are represented by right quasigroup mappings
gap> IsomorphismRightQuasigroups(AsLoop(G),S);
MappingByFunction( <associative loop of size 21>, \ 
<Moufang loop of size 21>, function( x ) ... end )