457 lines
14 KiB
Plaintext
457 lines
14 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 a general principle: collections from finite families are finite.
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InstallMethod(IsFinite, "for any collection (with a finite element family)",
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[IsCollection],
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function(C)
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local ef;
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ef := ElementsFamily(FamilyObj(C));
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if HasIsFinite(ef) and IsFinite(ef) then return true; fi;
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TryNextMethod();
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return fail;
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end);
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## And now create them from multiplication tables
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# First a helper function
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MagmaNumber@ := 1;
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MagmaLetters@ := "VABCDEFGHJKMNPSTU";
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MagmaBase@ := Length(MagmaLetters@);
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NextMagmaString@ := function(filts)
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local str, n;
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str := "l";
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if not "IsLeftQuotientElement" in NamesFilter(filts) then
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str := "r";
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fi;
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n := MagmaNumber@;
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MagmaNumber@ := MagmaNumber@ + 1;
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while n > 0 do
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Add(str, MagmaLetters@[RemInt(n, MagmaBase@) + 1]);
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n := QuoInt(n, MagmaBase@);
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od;
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return str;
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end;
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FiniteMagmaCreator@ := function(tbl, cnstr, filts)
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local M;
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M := MagmaByMultiplicationTableCreatorNC(
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tbl, cnstr, filts and IsMagmaByMultiplicationTableObj);
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# Is there such a thing as a non-finite table in GAP? Anyhow...
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SetIsFinite(ElementsFamily(FamilyObj(M)), IsFinite(tbl));
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SpecifyElmNamePrefix(M, NextMagmaString@(filts));
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return M;
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end;
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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 -> FiniteMagmaCreator@(T, LeftQuasigroupNC, IsLeftQuotientElement)
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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 -> FiniteMagmaCreator@(T, RightQuasigroupNC, IsRightQuotientElement)
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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 -> FiniteMagmaCreator@(T, LeftRackNC,
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IsLeftQuotientElement and IsLSelfDistElement)
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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 -> FiniteMagmaCreator@(T, LeftQuandleNC,
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IsLeftQuotientElement and IsLSelfDistElement and IsIdempotent)
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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 -> FiniteMagmaCreator@(T, RightRackNC,
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IsRightQuotientElement and IsRSelfDistElement)
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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 -> FiniteMagmaCreator@(T, RightQuandleNC,
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IsRightQuotientElement and IsRSelfDistElement and IsIdempotent)
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);
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## Creators from permutations
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InstallGlobalFunction(LeftQuasigroupByPerms,
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function(perms)
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local Q;
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Q := LeftQuasigroupByMultiplicationTableNC(
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CayleyTableByPerms(perms, [1..Length(perms)]));
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SetLeftPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(LeftRackByPerms,
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function(perms)
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local Q;
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Q := LeftRackByMultiplicationTable(
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CayleyTableByPerms(perms, [1..Length(perms)]));
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SetLeftPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(LeftRackByPermsNC,
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function(perms)
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local Q;
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Q := LeftRackByMultiplicationTableNC(
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CayleyTableByPerms(perms, [1..Length(perms)]));
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SetLeftPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(LeftQuandleByPerms,
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function(perms)
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local Q;
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Q := LeftQuandleByMultiplicationTable(
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CayleyTableByPerms(perms, [1..Length(perms)]));
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SetLeftPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(LeftQuandleByPermsNC,
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function(perms)
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local Q;
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Q := LeftQuandleByMultiplicationTableNC(
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CayleyTableByPerms(perms, [1..Length(perms)]));
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SetLeftPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(RightQuasigroupByPerms,
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function(perms)
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local Q;
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Q := RightQuasigroupByMultiplicationTableNC(
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TransposedMat(
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CayleyTableByPerms(perms, [1..Length(perms)])));
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SetRightPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(RightRackByPerms,
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function(perms)
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local Q;
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Q := RightRackByMultiplicationTable(
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TransposedMat(
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CayleyTableByPerms(perms, [1..Length(perms)])));
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SetRightPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(RightRackByPermsNC,
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function(perms)
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local Q;
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Q := RightRackByMultiplicationTableNC(
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TransposedMat(
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CayleyTableByPerms(perms, [1..Length(perms)])));
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SetRightPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(RightQuandleByPerms,
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function(perms)
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local Q;
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Q := RightQuandleByMultiplicationTable(
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TransposedMat(
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CayleyTableByPerms(perms, [1..Length(perms)])));
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SetRightPerms(Q, perms);
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return Q;
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end);
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InstallGlobalFunction(RightQuandleByPermsNC,
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function(perms)
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local Q;
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Q := RightQuandleByMultiplicationTableNC(
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TransposedMat(
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CayleyTableByPerms(perms, [1..Length(perms)])));
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SetRightPerms(Q, perms);
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return Q;
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end);
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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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InstallImmediateMethod(IsBuiltFromMultiplicationTable,
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IsCollection, 1,
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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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# Note that opposite of quandles seems to come for free from above, which is
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# good.
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## Direct products
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# As in the case of general one-sided quasigroups, this implementation should
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# only be called with a second-argument collection which is not a group or
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# quasigroup.
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InstallOtherMethod(DirectProductOp,
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"for a list and non-quasigroup magma built from multiplication table",
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[IsList, IsMagma and IsBuiltFromMultiplicationTable],
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function (list, first)
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local n, item, i, jof, bigtable;
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n := Length(list);
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# Simple checks
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if n < 1 then
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Error("Usage: Cannot take DirectProduct of zero items.");
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elif n < 2 then
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return list[1];
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fi;
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# See if we can handle all objects
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for i in [2..n] do
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if not HasMultiplicationTable(list[i]) then
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return DirectProductOp(Permuted(list, (1,i)), list[i]);
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fi;
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od;
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# OK, safe to take everyone's multiplication table.
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# So go ahead and make the big thing
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bigtable := ProductTableOfCanonicalCayleyTables(
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List(list, MultiplicationTable));
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# But we have to figure out what to do with it.
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jof := RoughJoinOfFilters@(list, first);
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# Dispatch modeled after the general one
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if "IsMagmaWithOne" in jof then
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if "IsMonoid" in jof then
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return MonoidByMultiplicationTable(bigtable);
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fi;
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return MagmaWithOneByMultiplicationTable(bigtable);
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fi;
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if "IsAssociative" in jof then
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return SemigroupByMultiplicationTable(bigtable);
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elif "IsLeftQuasigroup" in jof then
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if "IsLSelfDistributive" in jof then
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if "IsElementwiseIdempotent" in jof then
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return LeftQuandleByMultiplicationTableNC(bigtable);
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fi;
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return LeftRackByMultiplicationTableNC(bigtable);
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fi;
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return LeftQuasigroupByMultiplicationTableNC(bigtable);
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elif "IsRightQuasigroup" in jof then
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if "IsRSelfDistributive" in jof then
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if "IsElementwiseIdempotent" in jof then
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return RightQuandleByMultiplicationTableNC(bigtable);
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fi;
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return RightRackByMultiplicationTableNC(bigtable);
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fi;
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return RightQuasigroupByMultiplicationTableNC(bigtable);
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fi;
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return MagmaByMultiplicationTable(bigtable);
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end);
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