267 lines
8.9 KiB
Plaintext
267 lines
8.9 KiB
Plaintext
## bytable.gi RAQ Implementation of racks etc. by multiplication tables.
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## Predicates to check tables for distributivity
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InstallMethod(IsRightSelfDistributiveTable, "for matrix",
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[ IsMatrix ],
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T -> IsLeftSelfDistributiveTable(TransposedMat(T))
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);
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InstallMethod(IsLeftSelfDistributiveTable, "for matrix",
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[ IsMatrix ],
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function(T)
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# Everybody else does it by checking all of the cases, so why not me, too?
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# Is there a better way?
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local n,i,j,k;
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n := Length(T);
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for i in [1..n] do for j in [1..n] do for k in [1..n] do
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if T[i, T[j,k]] <> T[T[i,j], T[i,k]] then return false; fi;
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od; od; od;
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return true;
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end);
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InstallMethod(IsElementwiseIdempotentTable, "for matrix",
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[ IsMatrix ],
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T -> ForAll([1..Length(T)], i->(T[i,i]=i))
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);
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## And now create them from multiplication tables
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InstallGlobalFunction(LeftQuasigroupByMultiplicationTable,
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function(T)
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if not IsLeftQuasigroupTable(T) then
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Error("Multiplication table <T> must have each row a permutation of ",
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"the same entries.");
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fi;
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return LeftQuasigroupByMultiplicationTableNC(
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CanonicalCayleyTableOfLeftQuasigroupTable(T));
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end);
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InstallGlobalFunction(LeftQuasigroupByMultiplicationTableNC,
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T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuasigroupNC,
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IsLeftQuotientElement and IsMagmaByMultiplicationTableObj)
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);
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InstallGlobalFunction(RightQuasigroupByMultiplicationTable,
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function(T)
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if not IsRightQuasigroupTable(T) then
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Error("Multiplication table <T> must have each row a permutation of ",
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"the same entries.");
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fi;
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return RightQuasigroupByMultiplicationTableNC(CanonicalCayleyTable(T));
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end);
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InstallGlobalFunction(RightQuasigroupByMultiplicationTableNC,
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T -> MagmaByMultiplicationTableCreatorNC(T, RightQuasigroupNC,
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IsRightQuotientElement and IsMagmaByMultiplicationTableObj)
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);
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InstallGlobalFunction(LeftRackByMultiplicationTable,
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function(T)
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if not IsLeftQuasigroupTable(T) then
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Error("Multiplication table <T> must have each row a permutation of ",
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"the same entries.");
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fi;
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T := CanonicalCayleyTableOfLeftQuasigroupTable(T);
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if not IsLeftSelfDistributiveTable(T) then
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Error("Multiplication table <T> must be left self distributive.");
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fi;
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return LeftRackByMultiplicationTableNC(T);
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end);
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InstallGlobalFunction(LeftRackByMultiplicationTableNC,
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T -> MagmaByMultiplicationTableCreatorNC(T, LeftRackNC,
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IsLeftQuotientElement and IsLSelfDistElement and
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IsMagmaByMultiplicationTableObj
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)
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);
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InstallGlobalFunction(LeftQuandleByMultiplicationTable,
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function(T)
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if not IsLeftQuasigroupTable(T) then
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Error("Multiplication table <T> must have each row a permutation of ",
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"the same entries.");
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fi;
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T := CanonicalCayleyTableOfLeftQuasigroupTable(T);
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if not (IsLeftSelfDistributiveTable(T) and
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IsElementwiseIdempotentTable(T)) then
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Error("Multiplication table <T> must be left self-dist and idempotent.");
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fi;
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return LeftQuandleByMultiplicationTableNC(T);
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end);
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InstallGlobalFunction(LeftQuandleByMultiplicationTableNC,
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T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuandleNC,
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IsLeftQuotientElement and IsLSelfDistElement and IsIdempotent and
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IsMagmaByMultiplicationTableObj
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)
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);
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InstallGlobalFunction(RightRackByMultiplicationTable,
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function(T)
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if not IsRightQuasigroupTable(T) then
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Error("Multiplication table <T> must have each column a permutation of ",
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"the same entries.");
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fi;
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T := CanonicalCayleyTable(T);
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if not IsRightSelfDistributiveTable(T) then
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Error("Multiplication table <T> must be right self distributive.");
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fi;
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return RightRackByMultiplicationTableNC(T);
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end);
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InstallGlobalFunction(RightRackByMultiplicationTableNC,
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T -> MagmaByMultiplicationTableCreatorNC(T, RightRackNC,
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IsRightQuotientElement and IsRSelfDistElement and
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IsMagmaByMultiplicationTableObj
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)
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);
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InstallGlobalFunction(RightQuandleByMultiplicationTable,
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function(T)
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if not IsRightQuasigroupTable(T) then
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Error("Multiplication table <T> must have each column a permutation of ",
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"the same entries.");
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fi;
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T := CanonicalCayleyTable(T);
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if not (IsRightSelfDistributiveTable(T) and
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IsElementwiseIdempotentTable(T)) then
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Error("Multiplication table <T> must be right self-dist and idempotent.");
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fi;
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return RightQuandleByMultiplicationTableNC(T);
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end);
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InstallGlobalFunction(RightQuandleByMultiplicationTableNC,
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T -> MagmaByMultiplicationTableCreatorNC(T, RightQuandleNC,
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IsRightQuotientElement and IsRSelfDistElement and IsIdempotent and
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IsMagmaByMultiplicationTableObj
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)
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);
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## And define the operations
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InstallOtherMethod(LeftQuotient,
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"for two elts in magma by mult table, when left has left quotients",
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IsIdenticalObj,
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[IsLeftQuotientElement, IsMagmaByMultiplicationTableObj],
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function (l,r)
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local fam, ix;
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fam := FamilyObj(l);
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ix := LeftDivisionTable(fam)[l![1],r![1]];
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return fam!.set[ix];
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end);
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InstallOtherMethod(\/,
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"for two elts in magma by mult table, when right has right quotients",
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IsIdenticalObj,
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[IsMagmaByMultiplicationTableObj, IsRightQuotientElement],
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function (l,r)
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local fam, ix;
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fam := FamilyObj(r);
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ix := RightDivisionTable(fam)[l![1],r![1]];
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return fam!.set[ix];
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end);
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## Create division tables as needed
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InstallMethod(LeftDivisionTable,
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"for an object with a multiplication table",
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[HasMultiplicationTable],
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function(fam)
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local LS, n;
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LS := LeftPerms(fam);
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n := Size(LS);
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return List(LS, x->ListPerm(Inverse(x), n));
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end);
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InstallMethod(RightDivisionTable,
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"for an object with a multiplication table",
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[HasMultiplicationTable],
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function (obj)
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local RS, n;
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RS := RightPerms(obj);
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n := Size(RS);
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return TransposedMat(List(RS, x->ListPerm(Inverse(x), n)));
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end);
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## Create perm lists as needed
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InstallMethod(LeftPerms,
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"for an object with a multiplication table",
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[HasMultiplicationTable],
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function(fam)
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return List(MultiplicationTable(fam), x->PermList(x));
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end);
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InstallMethod(RightPerms,
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"for an object with a muliplication table",
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[HasMultiplicationTable],
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function(fam)
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return List(TransposedMat(MultiplicationTable(fam)), x->PermList(x));
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end);
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## Distributivity/idempotence checkers for when need be
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InstallMethod(IsLSelfDistributive, "for collections with multiplication tables",
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[IsMultiplicativeElementCollection and HasMultiplicationTable],
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M -> IsLeftSelfDistributiveTable(MultiplicationTable(M))
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);
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InstallMethod(IsRSelfDistributive, "for collections with multiplication table",
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[IsMultiplicativeElementCollection and HasMultiplicationTable],
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M -> IsRightSelfDistributiveTable(MultiplicationTable(M))
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);
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## Patch View/Print/Display for magma by mult objects
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InstallMethod(String, "for an element of magma by multiplication table",
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[IsMagmaByMultiplicationTableObj],
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function(obj)
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local fam;
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fam := FamilyObj(obj);
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if IsBound(fam!.elmNamePrefix) then
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return Concatenation(fam!.elmNamePrefix, String(obj![1]));
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fi;
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return Concatenation("m", String(obj![1]));
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end);
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InstallMethod(ViewString, "for an element of magma by multiplication table",
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[IsMagmaByMultiplicationTableObj],
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obj -> String(obj));
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InstallMethod(DisplayString, "for an element of magma by multiplication table",
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[IsMagmaByMultiplicationTableObj],
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obj -> String(obj));
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InstallMethod(PrintObj, "for an element of magma by multiplication table",
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[IsMagmaByMultiplicationTableObj],
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function(obj) Print(String(obj)); end);
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## Property of a collection that its elements know their multiplication table
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InstallMethod(IsBuiltFromMultiplicationTable, "for a collection",
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[IsCollection],
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C -> HasMultiplicationTable(ElementsFamily(FamilyObj(C)))
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);
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## Special case the Opposite function from LOOPS package, since the opposite
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## of a left quasigroup is a right quasigroup and vice versa
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# Is there a way to do this just once for each direction?
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InstallMethod(Opposite, "for left quasigroup",
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[ IsLeftQuasigroup and IsBuiltFromMultiplicationTable],
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L -> RightQuasigroupByMultiplicationTable(
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TransposedMat(MultiplicationTable(L))
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)
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);
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InstallMethod(Opposite, "for left rack",
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[ IsLeftRack and IsBuiltFromMultiplicationTable],
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L -> RightRackByMultiplicationTableNC(TransposedMat(MultiplicationTable(L)))
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);
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InstallMethod(Opposite, "for right quasigroup",
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[ IsRightQuasigroup and IsBuiltFromMultiplicationTable],
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L -> LeftQuasigroupByMultiplicationTable(
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TransposedMat(MultiplicationTable(L))
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)
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);
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InstallMethod(Opposite, "for right rack",
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[ IsRightRack and IsBuiltFromMultiplicationTable],
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L -> LeftRackByMultiplicationTableNC(TransposedMat(MultiplicationTable(L)))
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);
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