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

## Create structures with generators
InstallGlobalFunction(CloneOfTypeByGenerators,
  function(cat, fam, gens, genAttrib, tableCstr)
    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));
    SetConstructorFromTable(M, tableCstr);
    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);
  fi;
  return CloneOfTypeByGenerators(IsLeftQuasigroup, fam, arg,
                                 GeneratorsOfLeftQuasigroup, 
                                 LeftQuasigroupByMultiplicationTableNC);
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);
  fi;
  return CloneOfTypeByGenerators(IsLeftRack, fam, arg,
                                 GeneratorsOfLeftQuasigroup,
                                 LeftRackByMultiplicationTableNC);
end);

InstallGlobalFunction(LeftQuandle, function(arg)
  local fam;
  if Length(arg) = 0 then
    Error("usage: LeftQuandle([<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);
  fi;
  return CloneOfTypeByGenerators(IsLeftQuandle, fam, arg,
                                 GeneratorsOfLeftQuasigroup,
                                 LeftQuandleByMultiplicationTableNC);
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);
  fi;
  return CloneOfTypeByGenerators(IsRightQuasigroup, fam, arg,
                                 GeneratorsOfRightQuasigroup,
                                 RightQuasigroupByMultiplicationTableNC);
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);
  fi;
  return CloneOfTypeByGenerators(IsRightRack, fam, arg,
                                 GeneratorsOfRightQuasigroup,
                                 RightRackByMultiplicationTableNC);
end);

InstallGlobalFunction(RightQuandle, function(arg)
  local fam;
  if Length(arg) = 0 then
    Error("usage: RightQuandle([<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);
  fi;
  return CloneOfTypeByGenerators(IsRightQuandle, fam, arg,
                                 GeneratorsOfRightQuasigroup,
                                 RightQuandleByMultiplicationTableNC);
end);


## View and print and such
LeftObjString@ := function(Q)
  # Don't test distributivity if we haven't already
  if HasIsLSelfDistributive(Q) and IsLeftRack(Q) then 
    if HasIsElementwiseIdempotent(Q) and IsElementwiseIdempotent(Q) then
      return "LeftQuandle";
    fi;
    return "LeftRack";
  fi;
  return "LeftQuasigroup";
end;

RightObjString@ := function(Q)
  # Don't test distributivity if we haven't already
  if HasIsRSelfDistributive(Q) and IsRightRack(Q) then 
    if HasIsElementwiseIdempotent(Q) and IsElementwiseIdempotent(Q) then
      return "RightQuandle";
    fi;
    return "RightRack";
  fi;
  return "RightQuasigroup";
end;

InstallMethod(String, "for a left quasigroup",
              [IsLeftQuasigroup],
              Q -> Concatenation(LeftObjString@(Q), "(...)"));

InstallMethod(String, "for a left quasigroup with generators",
              [IsLeftQuasigroup and HasGeneratorsOfLeftQuasigroup],
              Q -> Concatenation(LeftObjString@(Q), "( ", 
                                 String(GeneratorsOfLeftQuasigroup(Q)), " )"));

InstallMethod(String, "for a left quasigroup with multiplication table",
              [IsLeftQuasigroup and HasMultiplicationTable],
              Q -> Concatenation(LeftObjString@(Q), 
                                 "ByMultiplicationTableNC( ",
                                 String(MultiplicationTable(Q)), " )"));

InstallMethod(String, "for a right quasigroup",
              [IsRightQuasigroup],
              Q -> Concatenation(RightObjString@(Q), "(...)"));

InstallMethod(String, "for a right quasigroup with generators",
              [IsRightQuasigroup and HasGeneratorsOfRightQuasigroup],
              Q -> Concatenation(RightObjString@(Q), "( ", 
                                 String(GeneratorsOfRightQuasigroup(Q)), " )"));

InstallMethod(String, "for a right quasigroup with multiplication table",
              [IsRightQuasigroup and HasMultiplicationTable],
              Q -> Concatenation(RightObjString@(Q), 
                                 "ByMultiplicationTableNC( ",
                                 String(MultiplicationTable(Q)), " )"));

InstallMethod(PrintString, "for a left quasigroup",
              [IsLeftQuasigroup], Q -> String(Q));

InstallMethod(PrintString, "for a right quasigroup",
              [IsRightQuasigroup], Q -> String(Q));

InstallMethod(DisplayString, "for a left quasigroup",
              [IsLeftQuasigroup], Q -> String(Q));

InstallMethod(DisplayString, "for a right quasigroup",
              [IsRightQuasigroup], Q -> String(Q));

InstallMethod(Display, "for a left quasigroup with multiplication table",
  [IsLeftQuasigroup and HasMultiplicationTable],
  function(Q)
    Print(LeftObjString@(Q), " with ", Size(Q), 
          " elements, generated by ", 
          GeneratorsOfLeftQuasigroup(Q), ", with table\n");
    Display(MultiplicationTable(Q));
end);

InstallMethod(Display, "for a right quasigroup with multiplication table",
  [IsRightQuasigroup and HasMultiplicationTable],
  function(Q)
    Print(RightObjString@(Q), " with ", Size(Q), 
          " elements, generated by ", 
          GeneratorsOfRightQuasigroup(Q), ", with table\n");
    Display(MultiplicationTable(Q));
end);

LeftObjView@ := function(Q)
  # Don't test distributivity if we haven't already
  if HasIsLSelfDistributive(Q) and IsLeftRack(Q) then 
    if HasIsElementwiseIdempotent(Q) and IsElementwiseIdempotent(Q) then
      return "<left quandle";
    fi;
    return "<left rack";
  fi;
  return "<left quasigroup";
end;

RightObjView@ := function(Q)
  # Don't test distributivity if we haven't already
  if HasIsRSelfDistributive(Q) and IsRightRack(Q) then 
    if HasIsElementwiseIdempotent(Q) and IsElementwiseIdempotent(Q) then
      return "<right quandle";
    fi;
    return "<right rack";
  fi;
  return "<right quasigroup";
end;

InstallMethod(ViewString, "for a left quasigroup",
              [IsLeftQuasigroup], 
              Q -> Concatenation(LeftObjView@(Q), ">"));

InstallMethod(ViewString, "for a left quasigroup with generators",
              [IsLeftQuasigroup and HasGeneratorsOfLeftQuasigroup], 
              Q -> Concatenation(LeftObjView@(Q), " with ",
                                 String(Size(GeneratorsOfLeftQuasigroup(Q))), 
                                 " generators>"));

InstallMethod(ViewString, "for a right quasigroup",
              [IsRightQuasigroup], 
              Q -> Concatenation(RightObjView@(Q), ">"));

InstallMethod(ViewString, "for a right quasigroup with generators",
              [IsRightQuasigroup and HasGeneratorsOfRightQuasigroup], 
              Q -> Concatenation(RightObjView@(Q), " with ",
                                 String(Size(GeneratorsOfRightQuasigroup(Q))), 
                                 " generators>"));