glen/RAQ
1
0
Fork 0
Code Issues 1 Pull Requests Releases Wiki Activity

Check that putative generators of a structure at least satisfy the structure axioms

Browse Source
This commit is contained in:
Glen Whitney 2017-10-20 15:25:22 +02:00
parent 3c4eb2582a
commit 7d1ff6c683
3 changed files with 144 additions and 106 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

25
gap/bytable.gi
View File

@ -36,7 +36,7 @@ InstallGlobalFunction(LeftQuasigroupByMultiplicationTable,
end);
InstallGlobalFunction(LeftQuasigroupByMultiplicationTableNC,
T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuasigroup,
T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuasigroupNC,
IsLeftQuotientElement and IsMagmaByMultiplicationTableObj)
);
@ -50,7 +50,7 @@ InstallGlobalFunction(RightQuasigroupByMultiplicationTable,
end);
InstallGlobalFunction(RightQuasigroupByMultiplicationTableNC,
T -> MagmaByMultiplicationTableCreatorNC(T, RightQuasigroup,
T -> MagmaByMultiplicationTableCreatorNC(T, RightQuasigroupNC,
IsRightQuotientElement and IsMagmaByMultiplicationTableObj)
);
@ -68,7 +68,7 @@ InstallGlobalFunction(LeftRackByMultiplicationTable,
end);
InstallGlobalFunction(LeftRackByMultiplicationTableNC,
T -> MagmaByMultiplicationTableCreatorNC(T, LeftRack,
T -> MagmaByMultiplicationTableCreatorNC(T, LeftRackNC,
IsLeftQuotientElement and IsLSelfDistElement and
IsMagmaByMultiplicationTableObj
)
@ -89,7 +89,7 @@ InstallGlobalFunction(LeftQuandleByMultiplicationTable,
end);
InstallGlobalFunction(LeftQuandleByMultiplicationTableNC,
T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuandle,
T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuandleNC,
IsLeftQuotientElement and IsLSelfDistElement and IsIdempotent and
IsMagmaByMultiplicationTableObj
)
@ -109,7 +109,7 @@ InstallGlobalFunction(RightRackByMultiplicationTable,
end);
InstallGlobalFunction(RightRackByMultiplicationTableNC,
T -> MagmaByMultiplicationTableCreatorNC(T, RightRack,
T -> MagmaByMultiplicationTableCreatorNC(T, RightRackNC,
IsRightQuotientElement and IsRSelfDistElement and
IsMagmaByMultiplicationTableObj
)
@ -130,7 +130,7 @@ InstallGlobalFunction(RightQuandleByMultiplicationTable,
end);
InstallGlobalFunction(RightQuandleByMultiplicationTableNC,
T -> MagmaByMultiplicationTableCreatorNC(T, RightQuandle,
T -> MagmaByMultiplicationTableCreatorNC(T, RightQuandleNC,
IsRightQuotientElement and IsRSelfDistElement and IsIdempotent and
IsMagmaByMultiplicationTableObj
)
@ -197,21 +197,16 @@ InstallMethod(RightPerms,
end);
## Distributivity/idempotence checkers for when need be
InstallMethod(IsLSelfDistributive, "for magma",
[IsMagma and IsFinite],
InstallMethod(IsLSelfDistributive, "for collections with multiplication tables",
[HasMultiplicationTable],
M -> IsLeftSelfDistributiveTable(MultiplicationTable(M))
);
InstallMethod(IsRSelfDistributive, "for magma",
[IsMagma and IsFinite],
InstallMethod(IsRSelfDistributive, "for collections with multiplication table",
[HasMultiplicationTable],
M -> IsRightSelfDistributiveTable(MultiplicationTable(M))
);
InstallMethod(IsElementwiseIdempotent, "for magma",
[IsMagma and IsFinite],
M -> ForAll(Elements(M), m->IsIdempotent(m))
);
## Patch View/Print/Display for magma by mult objects
InstallMethod(String, "for an element of magma by multiplication table",
[IsMagmaByMultiplicationTableObj],

26
gap/structure.gd
View File

@ -21,15 +21,13 @@ DeclareCategoryCollections("IsLSelfDistElement");
DeclareCategory("IsRSelfDistElement", IsMultiplicativeElement);
DeclareCategoryCollections("IsRSelfDistElement");
# Left self-distributive magmas:
DeclareProperty("IsLSelfDistributive", IsMagma);
InstallTrueMethod(IsLSelfDistributive,
IsMagma and IsLSelfDistElementCollection);
# Left self-distributive collections of elements:
DeclareProperty("IsLSelfDistributive", IsMultiplicativeElementCollection);
InstallTrueMethod(IsLSelfDistributive, IsLSelfDistElementCollection);
# Right self-distributive magmas:
DeclareProperty("IsRSelfDistributive", IsMagma);
InstallTrueMethod(IsRSelfDistributive,
IsMagma and IsRSelfDistElementCollection);
# Right self-distributive collections of elements:
DeclareProperty("IsRSelfDistributive", IsMultiplicativeElementCollection);
InstallTrueMethod(IsRSelfDistributive, IsRSelfDistElementCollection);
## Idempotence
# There is already a property IsIdempotent on elements, but to definw
@ -37,9 +35,9 @@ InstallTrueMethod(IsRSelfDistributive,
# collections category:
DeclareCategoryCollections("IsIdempotent");
# Idempotent magmas, i.e. magmas in which every element is idempotent
DeclareProperty("IsElementwiseIdempotent", IsMagma);
InstallTrueMethod(IsElementwiseIdempotent, IsMagma and IsIdempotentCollection);
# Collections in which every element is idempotent
DeclareProperty("IsElementwiseIdempotent", IsMultiplicativeElementCollection);
InstallTrueMethod(IsElementwiseIdempotent, IsIdempotentCollection);
## Left and right racks and quandles
DeclareSynonym("IsLeftRack", IsLeftQuasigroup and IsLSelfDistributive);
@ -54,11 +52,17 @@ DeclareSynonym("IsRightQuandle", IsLeftRack and IsElementwiseIdempotent);
# the family of elements of M may be specified, and must be if <gens>
# is empty (in which case M will be empty as well).
DeclareGlobalFunction("LeftQuasigroup");
DeclareGlobalFunction("LeftQuasigroupNC");
DeclareGlobalFunction("RightQuasigroup");
DeclareGlobalFunction("RightQuasigroupNC");
DeclareGlobalFunction("LeftRack");
DeclareGlobalFunction("LeftRackNC");
DeclareGlobalFunction("RightRack");
DeclareGlobalFunction("RightRackNC");
DeclareGlobalFunction("LeftQuandle");
DeclareGlobalFunction("LeftQuandleNC");
DeclareGlobalFunction("RightQuandle");
DeclareGlobalFunction("RightQuandleNC");
# Underlying operation
DeclareGlobalFunction("CloneOfTypeByGenerators");

197
gap/structure.gi
View File

@ -1,6 +1,36 @@
## structure.gi RAQ Implementation of definitiions, reps, and elt operations
## Testing properties of collections the hard way if we have to
InstallMethod(IsElementwiseIdempotent, "for finite collections",
[IsMultiplicativeElementCollection and IsFinite],
M -> ForAll(Elements(M), m->IsIdempotent(m))
);
InstallMethod(IsLSelfDistributive,
"for arbitrary multiplicative collections, the hard way",
[IsMultiplicativeElementCollection],
function (C)
local a,b,d;
for a in C do for b in C do for d in C do
if d*(a*b) <> (d*a)*(d*b) then return false; fi;
od; od; od;
return true;
end);
InstallMethod(IsRSelfDistributive,
"for arbitrary multiplicative collections, the hard way",
[IsMultiplicativeElementCollection],
function (C)
local a,b,d;
for a in C do for b in C do for d in C do
if (a*b)*d <> (a*d)*(b*d) then return false; fi;
od; od; od;
return true;
end);
## Create structures with generators
InstallGlobalFunction(CloneOfTypeByGenerators,
function(cat, fam, gens, genAttrib, tableCstr)
local M;
@ -13,122 +43,131 @@ InstallGlobalFunction(CloneOfTypeByGenerators,
return M;
end);
## Helpers for the constructors below:
ArgHelper@ := function(parmlist)
# returns a list of the family and the flat list of elements of parmlist
if Length(parmlist) = 0 then
Error("usage: RAQ constructors take an optional family, followed by gens");
fi;
if IsFamily(parmlist[1]) then return [Remove(parmlist,1), Flat(parmlist)]; fi;
parmlist := Flat(parmlist);
return [FamilyObj(parmlist), parmlist];
end;
CheckLQGprop@ := function(gens)
local g, h;
# Make sure all elements in gens have left quotient property pairwise
for g in gens do for h in gens do
if g*LeftQuotient(g,h) <> h or LeftQuotient(g,g*h) <> h then
Error("left quasigroup property of left quotients violated");
fi;
od; od;
return;
end;
CheckRQGprop@ := function(gens)
local g, h;
# Make sure all elements in gens have right quotient property pairwise
for g in gens do for h in gens do
if (h*g)/g <> h or (h/g)*g <> h then
Error("right quasigroup property of / violated");
fi;
od; od;
return;
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,
arg := ArgHelper@(arg);
CheckLQGprop@(arg[2]);
return LeftQuasigroupNC(arg[1], arg[2]);
end);
InstallGlobalFunction(LeftQuasigroupNC, function(fam, gens)
return CloneOfTypeByGenerators(IsLeftQuasigroup, fam, gens,
GeneratorsOfLeftQuasigroup,
LeftQuasigroupByMultiplicationTableNC);
end);
InstallGlobalFunction(LeftRack, function(arg)
local fam;
if Length(arg) = 0 then
Error("usage: LeftRack([<family>], <gens>)");
arg := ArgHelper@(arg);
CheckLQGprop@(arg[2]);
if not IsLSelfDistributive(arg[2]) then
Error("Left rack must have left distributive generators");
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,
return LeftRackNC(arg[1], arg[2]);
end);
IntallGlobalFunction(LeftRackNC, function(fam, gens)
return CloneOfTypeByGenerators(IsLeftRack, fam, gens,
GeneratorsOfLeftQuasigroup,
LeftRackByMultiplicationTableNC);
end);
InstallGlobalFunction(LeftQuandle, function(arg)
local fam;
if Length(arg) = 0 then
Error("usage: LeftQuandle([<family>], <gens>)");
arg := ArgHelper@(arg);
CheckLQGprop@(arg[2]);
if not IsLSelfDistributive(arg[2]) then
Error("Left quandle must have left distributive generators");
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);
if not IsElementwiseIdempotent(arg[2]) then
Error("Quandles must contain only idempotent elements");
fi;
return CloneOfTypeByGenerators(IsLeftQuandle, fam, arg,
return LeftQuandleNC(arg[1], arg[2]);
end);
InstallGlobalFunction(LeftQuandleNC, function(fam, gens)
return CloneOfTypeByGenerators(IsLeftQuandle, fam, gens,
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,
arg := ArgHelper@(arg);
CheckRQGprop@(arg[2]);
return RightQuasigroupNC(arg[1], arg[2]);
end);
InstallGlobalFunction(RightQuasigroupNC, function(fam, gens)
return CloneOfTypeByGenerators(IsRightQuasigroup, fam, gens,
GeneratorsOfRightQuasigroup,
RightQuasigroupByMultiplicationTableNC);
end);
InstallGlobalFunction(RightRack, function(arg)
local fam;
if Length(arg) = 0 then
Error("usage: RightRack([<family>], <gens>)");
arg := ArgHelper@(arg);
CheckRQGprop@(arg[2]);
if not IsRSelfDistributive(arg[2]) then
Error("Right rack must have right distributive generators");
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,
return RightRackNC(arg[1], arg[2]);
end);
IntallGlobalFunction(RightRackNC, function(fam, gens)
return CloneOfTypeByGenerators(IsRightRack, fam, gens,
GeneratorsOfRightQuasigroup,
RightRackByMultiplicationTableNC);
end);
InstallGlobalFunction(RightQuandle, function(arg)
local fam;
if Length(arg) = 0 then
Error("usage: RightQuandle([<family>], <gens>)");
arg := ArgHelper@(arg);
CheckLQGprop@(arg[2]);
if not IsRSelfDistributive(arg[2]) then
Error("Right quandle must have right distributive generators");
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);
if not IsElementwiseIdempotent(arg[2]) then
Error("Quandles must contain only idempotent elements");
fi;
return CloneOfTypeByGenerators(IsRightQuandle, fam, arg,
return RightQuandleNC(arg[1], arg[2]);
end);
InstallGlobalFunction(RightQuandleNC, function(fam, gens)
return CloneOfTypeByGenerators(IsRightQuandle, fam, gens,
GeneratorsOfRightQuasigroup,
RightQuandleByMultiplicationTableNC);
end);
## View and print and such
LeftObjString@ := function(Q)
# Don't test distributivity if we haven't already