## structure.gi RAQ Implementation of definitiions, reps, and elt operations

## Create structures with generators
InstallMethod(CloneOfTypeByGenerators,
              "for generic category of magma, family, collection, gen attrib",
              [IsFilter, IsFamily, IsCollection, IsAttribute],
  function(cat, fam, gens, genAttrib)
    local M;
    if not(IsEmpty(gens) or IsIdenticalObj(FamilyObj(gens), fam)) then
      Error("<fam> and family of <gens> do not match");
    fi;
    M := Objectify(NewType( fam, cat and IsAttributeStoringRep), rec());
    Setter(genAttrib)(M, AsList(gens));
    return M;
end);

## Functions for each of the magma categories here
InstallGlobalFunction(LeftQuasigroup, function(arg)
  local fam;
  if Length(arg) = 0 then
    Error("usage: LeftQuasigroup([<family>], <gens>)");
  fi;
  # Extract the family
  if IsFamily(arg[1]) then
    fam = arg[1];
    Remove(arg, 1);
    arg = Flat(arg);
  else
    arg = Flat(arg);
    fam = FamilyObj(arg[1]);
  fi;
  return CloneOfTypeByGenerators(IsLeftQuasigroup, fam, arg,
                                 GeneratorsOfLeftQuasigroup);
end);

InstallGlobalFunction(LeftRack, function(arg)
  local fam;
  if Length(arg) = 0 then
    Error("usage: LeftRack([<family>], <gens>)");
  fi;
  # Extract the family
  if IsFamily(arg[1]) then
    fam = arg[1];
    Remove(arg, 1);
    arg = Flat(arg);
  else
    arg = Flat(arg);
    fam = FamilyObj(arg[1]);
  fi;
  return CloneOfTypeByGenerators(IsLeftRack, fam, arg,
                                 GeneratorsOfLeftQuasigroup);
end);

InstallGlobalFunction(RightQuasigroup, function(arg)
  local fam;
  if Length(arg) = 0 then
    Error("usage: RightQuasigroup([<family>], <gens>)");
  fi;
  # Extract the family
  if IsFamily(arg[1]) then
    fam = arg[1];
    Remove(arg, 1);
    arg = Flat(arg);
  else
    arg = Flat(arg);
    fam = FamilyObj(arg[1]);
  fi;
  return CloneOfTypeByGenerators(IsRightQuasigroup, fam, arg,
                                 GeneratorsOfRightQuasigroup);
end);

InstallGlobalFunction(RightRack, function(arg)
  local fam;
  if Length(arg) = 0 then
    Error("usage: RightRack([<family>], <gens>)");
  fi;
  # Extract the family
  if IsFamily(arg[1]) then
    fam = arg[1];
    Remove(arg, 1);
    arg = Flat(arg);
  else
    arg = Flat(arg);
    fam = FamilyObj(arg[1]);
  fi;
  return CloneOfTypeByGenerators(IsRightRack, fam, arg,
                                 GeneratorsOfRightQuasigroup);
end);

## And now create them from multiplication tables
InstallGlobalFunction(LeftQuasigroupByMultiplicationTable,
  function(T)
    if not IsLeftQuasigroupTable(T) then
      Error("Multiplication table <T> must have each row a permutation of ",
            "the same entries.");
    fi;
    return MagmaByMultiplicationTableCreatorNC(
             CanonicalCayleyTableOfLeftQuasigroupTable(T), 
             LeftQuasigroup,
             IsLeftQuotientElement and IsMagmaByMultiplicationTableObj
           );
end);

## And define the operation

InstallOtherMethod(LeftQuotient,
  "for two elts in magma by mult table, when left has left quotients",
  IsIdenticalObj,
  [IsLeftQuotientElement, IsMagmaByMultiplicationTableObj],
  function (l,r)
    return DivisionTable(FamilyObj(l))[l![1],r![1]];
end);

## Create division tables as needed
InstallMethod(LeftDivisionTable,
  "for a family with a multiplication table",
  [IsFamily and HasMultiplicationTable],
  function(fam)
    local LS, n;
    LS := LeftSectionList(fam);
    n := Size(T);
    return List(T, x->ListPerm(Inverse(x), n));
end);

## Create section lists as needed
InstallMethod(LeftSectionList,
  "for a family with a multiplication table",
  [IsFamily and HasMultiplicationTable],
  function(fam)
    return List(MultiplicationTable(fam), x->PermList(x));
end);