837 lines
29 KiB
Plaintext
837 lines
29 KiB
Plaintext
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#############################################################################
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##
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#W quasigroups.gi Representing, creating and displaying quasigroups [loops]
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##
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#H @(#)$Id: creation.gi, v 3.2.0 2015/11/22 gap Exp $
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##
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#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
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#Y P. Vojtechovsky (University of Denver, USA)
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##
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#############################################################################
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## TESTING MULTIPLICATION TABLES
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## -------------------------------------------------------------------------
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#############################################################################
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##
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#O IsQuasigroupTable( ls )
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##
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## Returns true if <ls> is an n by n latin square with n distinct
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## integral entries.
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InstallMethod( IsQuasigroupTable, "for matrix",
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[ IsMatrix ],
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function( ls )
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local first_row;
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# checking rows
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first_row := Set( ls[ 1 ] );
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if not Length( first_row ) = Length( ls[ 1 ] ) then return false; fi;
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if ForAll( ls, row -> Set( row ) = first_row ) = false then return false; fi;
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# checking columns
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ls := TransposedMat( ls );
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first_row := Set( ls[ 1 ] );
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if not Length( first_row ) = Length( ls[ 1 ] ) then return false; fi;
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if ForAll( ls, row -> Set( row ) = first_row ) = false then return false; fi;
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return true;
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end );
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#############################################################################
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##
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#O IsLoopTable( ls )
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##
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## Returns true if <ls> is a normalized latin square. An n by n latin square
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## is normalized if the first row and first column read the same, and the
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## entries in the first row are ordered.
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InstallMethod( IsLoopTable, "for matrix",
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[ IsMatrix ],
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function( ls )
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if not IsQuasigroupTable( ls ) then return false; fi;
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return Set( ls[ 1 ] ) = ls[ 1 ]
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and ls[ 1 ] = List( [1..Length(ls)], i -> ls[ i ][ 1 ] );
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end );
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#############################################################################
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##
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#O CanonicalCayleyTable( ls )
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##
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## Returns a Cayley table isomorphic to <ls>, in which the entries of ls
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## have been replaced by numerical values 1, ..., n in the following way:
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## Let e_1 < ... < e_n be all distinct entries of ls. Then e_i is renamed
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## to i. In particular, when {e_1,...e_n} = {1,...,n}, the operation
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## does nothing.
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InstallMethod( CanonicalCayleyTable, "for matrix",
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[ IsMatrix ],
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function( ls )
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local n, entries, i, j, T;
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n := Length( ls );
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# finding all distinct entries in the table
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entries := [];
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for i in [1..n] do for j in [1..n] do
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AddSet( entries, ls[ i ][ j ] );
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od; od;
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# renaming the entries and making a mutable copy, too
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T := List( [1..n], i -> [1..n] );
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for i in [1..n] do for j in [1..n] do
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T[ i ][ j ] := Position( entries, ls[ i ][ j ] );
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od; od;
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return T;
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end );
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#############################################################################
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##
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#O NormalizedQuasigroupTable( ls )
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##
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## Given a latin square <ls>, returns the corresponding normalized
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## latin square with entries 1, ..., n.
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InstallMethod( NormalizedQuasigroupTable, "for matrix",
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[ IsMatrix ],
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function( ls )
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local T, perm;
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if not IsQuasigroupTable( ls ) then
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Error( "LOOPS: <1> must be a latin square." );
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fi;
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# renaming the entries to be 1, ..., n
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T := CanonicalCayleyTable( ls );
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# permuting the columns so that the first row reads 1, ..., n
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perm := PermList( T[ 1 ] );
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T := List( T, row -> Permuted( row, perm ) );
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# permuting the rows so that the first column reads 1, ..., n
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return Set( T );
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end );
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#############################################################################
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## CREATING QUASIGROUPS AND LOOPS MANUALLY
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## -------------------------------------------------------------------------
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#############################################################################
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##
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#A CayleyTable( Q )
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##
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## Returns the Cayley table of the quasigroup <Q>
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InstallMethod( CayleyTable, "for quasigroup",
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[ IsQuasigroup ],
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function( Q )
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local elms, parent_elms;
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elms := Elements( Q );
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parent_elms := Elements( Parent( Q ) );
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return List( elms, x-> List( elms, y -> Position( parent_elms, x * y ) ) );
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end );
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#############################################################################
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##
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#O QuasigroupByCayleyTable( ct )
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##
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## Returns quasigroup with multiplication table <ct>.
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InstallMethod( QuasigroupByCayleyTable, "for matrix",
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[ IsMatrix ],
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function( ct )
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local F, Q, elms, n;
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if not IsQuasigroupTable( ct ) then
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Error( "LOOPS: <1> must be a latin square." );
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fi;
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# Making sure that entries are 1, ..., n
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ct := CanonicalCayleyTable( ct );
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# constructing the family
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F := NewFamily( "QuasigroupByCayleyTableFam", IsQuasigroupElement );
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# installing data for the family
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n := Length ( ct );
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F!.size := n;
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elms := Immutable( List( [1..n], i -> Objectify(
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NewType( F, IsQuasigroupElement and IsQuasigroupElmRep), [ i ] ) ) );
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F!.set := elms;
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F!.cayleyTable := ct;
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F!.names := "q";
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# creating the quasigroup
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Q := Objectify( NewType( FamilyObj( elms ),
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IsQuasigroup and IsAttributeStoringRep ), rec() );
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# setting attributes for the quasigroup
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SetSize( Q, n );
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SetAsSSortedList( Q, elms );
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SetParent( Q, Q );
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SetCayleyTable( Q, ct );
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return Q;
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end );
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#############################################################################
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##
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#O LoopByCayleyTable( ct )
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##
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## Returns loop with multiplication table <ct>.
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InstallMethod( LoopByCayleyTable, "for matrix",
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[ IsMatrix ],
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function( ct )
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local F, L, elms, n;
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if not IsLoopTable( ct ) then
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Error( "LOOPS: <1> must be a normalized latin square." );
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fi;
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# Making sure that the entries are 1, ..., n.
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# The table will remain normalized.
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ct := CanonicalCayleyTable( ct );
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# constructing the family
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F := NewFamily( "LoopByCayleyTableFam", IsLoopElement );
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# installing the data for the family
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n := Length ( ct );
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F!.size := n;
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elms := Immutable( List( [1..n], i -> Objectify(
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NewType( F, IsLoopElement and IsLoopElmRep), [ i ] ) ) );
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F!.set := elms;
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F!.cayleyTable := ct;
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F!.names := "l";
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# creating the loop
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L := Objectify( NewType( FamilyObj( elms ),
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IsLoop and IsAttributeStoringRep ), rec() );
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# setting attributes for the loop
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SetSize( L, n );
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SetAsSSortedList( L, elms );
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SetParent( L, L );
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SetCayleyTable( L, ct );
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SetOne( L, elms[ 1 ] );
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return L;
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end );
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#############################################################################
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##
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#O SetQuasigroupElmName( Q, name )
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##
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## Changes the name of elements of quasigroup or loop <Q> to <name>
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InstallMethod( SetQuasigroupElmName, "for quasigroup and string",
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[ IsQuasigroup, IsString ],
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function( Q, name )
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local F;
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F := FamilyObj( Elements( Q )[ 1 ] );
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F!.names := name;
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end);
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#############################################################################
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##
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#O CanonicalCopy( Q )
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##
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## Returns a canonical copy of <Q>, that is, an isomorphic object <O> with
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## canonical multiplication table and Parent( <O> ) = <O>.
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## (PROG) Properties and attributes are lost!
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InstallMethod( CanonicalCopy, "for quasigroup or loop",
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[ IsQuasigroup ],
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function( Q )
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if IsLoop( Q ) then
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return LoopByCayleyTable( CayleyTable( Q ) );
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fi;
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return QuasigroupByCayleyTable( CayleyTable( Q ) );
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end);
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#############################################################################
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## CREATING QUASIGROUPS AND LOOPS FROM A FILE
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## -------------------------------------------------------------------------
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#############################################################################
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##
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#F LOOPS_ReadCayleyTableFromFile( filename, replace_by_spaces )
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##
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## Auxiliary function. Reads the content of <filename> and tries to
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## interpret the data as a multiplication table, according to the rules
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## summarized below. If successful, it returns the multiplication table as
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## an n by n array.
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## ALGORITHM:
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## 1) the content of the file is read into one string
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## 2) all end-of-lines and all characters in the string <replace_by_spaces>
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## are replaced by spaces
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## 3) the string is split into chunks, where chunk = maximal substring
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## without spaces
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## 4) the number n of distinct chunks is found. If the number of chunks
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## is not n^2, error is announced and function terminates
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## 5) a numerical value 1 .. n is assigned to each chunk, depending on
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## its position among the distinct chunks
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## 6) multiplication table is constructed and returned
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InstallGlobalFunction( LOOPS_ReadCayleyTableFromFile,
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function( filename, replace_by_spaces)
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local input, s, i, chunks, started, starting_pos, z, j, distinct_chunks, c, n, T;
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if not ( IsString( filename) and IsString( replace_by_spaces ) ) then
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Error( "LOOPS: <1> must be a file name, and <2> must be a string." );
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fi;
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input := InputTextFile( filename );
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if input = fail then Error( "LOOPS: <1> is not a valid file name." ); fi;
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s := ReadAll( input );
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# removing end-of-lines, etc.
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for i in [1..Length( s )] do
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if (s[ i ] = '\n') or (s[ i ] in replace_by_spaces) then
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s[ i ] := ' ';
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fi;
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od;
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s[ Length( s ) + 1 ] := ' '; #to make sure that string ends with space
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#parsing string into chunks separated by spaces
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chunks := [];
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started := false; starting_pos := 0;
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for i in [1..Length( s )] do
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if not started then
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if not s[ i ] = ' ' then
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started := true;
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starting_pos := i;
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fi;
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else
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if s[ i ] = ' ' then #end of chunk
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z := "";
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for j in [ starting_pos..i-1 ] do
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z[ j - starting_pos + 1 ] := s[ j ];
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od;
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Add( chunks, z );
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started := false;
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fi;
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fi;
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od;
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distinct_chunks := [];
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for c in chunks do if not c in distinct_chunks then
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Add( distinct_chunks, c );
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fi; od;
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n := Length( distinct_chunks );
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if not Length( chunks ) = n^2 then
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Error( "LOOPS: The data in the file cannot be arranged into a square table." );
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fi;
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T := List( [1..n], i -> 0*[1..n] );
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for i in [1..n] do for j in [1..n] do
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T[ i ][ j ] := Position( distinct_chunks, chunks[ (i-1)*n + j ] );
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od; od;
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return T;
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end);
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#############################################################################
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##
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#O QuasigroupFromFile( filename, replace] )
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##
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## Calls LOOPS_ReadCayleyTableFromFile( filename, replace ) in order to return
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## the quasigroup with multiplication table in file <filename>.
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InstallMethod( QuasigroupFromFile, "for string and string",
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[ IsString, IsString ],
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function( filename, replace )
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return QuasigroupByCayleyTable( LOOPS_ReadCayleyTableFromFile( filename, replace ) );
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end );
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#############################################################################
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##
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#O LoopFromFile( filename , replace] )
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##
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## Calls LOOPS_ReadCayleyTableFromFile( filename, replace ) in order to return
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## the loop with multiplication table in file <filename>.
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InstallMethod( LoopFromFile, "for string and string",
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[ IsString, IsString ],
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function( filename, replace )
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return LoopByCayleyTable( LOOPS_ReadCayleyTableFromFile( filename, replace ) );
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end );
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#############################################################################
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## CREATING QUASIGROUPS AND LOOPS BY SECTIONS AND FOLDERS
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## -------------------------------------------------------------------------
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#############################################################################
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##
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#O CayleyTableByPerms( perms )
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##
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## Given a set <perms> of n permutations of an n-element set X, returns
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## n by n Cayley table ct such that ct[i][j] = X[j]^perms[i].
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## The operation is safe only if at most one permutation of <perms> is
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## the identity permutation, and all other permutations of <perms>
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## move all points of X.
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InstallMethod( CayleyTableByPerms,
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"for a list of permutations",
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[ IsPermCollection ],
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function( perms )
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local n, pts, max;
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n := Length( perms );
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if n=1 then
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return [ [ 1 ] ];
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fi;
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# one of perms[ 1 ], perms[ 2 ] must move all points
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pts := MovedPoints( perms[ 2 ] );
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if pts = [] then
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pts := MovedPoints( perms[ 1 ] );
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fi;
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max := Maximum( pts );
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|
|
# we permute the whole interval [1..max] and then keep only those coordinates corresponding to pts
|
||
|
|
return List( perms, p -> Permuted( [1..max], p^(-1) ){ pts } );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O QuasigroupByLeftSection( sect )
|
||
|
|
##
|
||
|
|
## Returns the quasigroup whose left section is the list of permutations
|
||
|
|
## <sect>.
|
||
|
|
|
||
|
|
InstallMethod( QuasigroupByLeftSection,
|
||
|
|
"for a set of left translation maps",
|
||
|
|
[ IsPermCollection ],
|
||
|
|
function( sect )
|
||
|
|
return QuasigroupByCayleyTable( CayleyTableByPerms( sect ) );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O LoopByLeftSection( sect )
|
||
|
|
##
|
||
|
|
## Returns the loop whose left section is the list of permutations <sect>.
|
||
|
|
## Since the order of translations in <sect> is determined by their
|
||
|
|
## image of the neutral element 1, we disregard the order.
|
||
|
|
|
||
|
|
InstallMethod( LoopByLeftSection,
|
||
|
|
"for a set of left translation maps",
|
||
|
|
[ IsPermCollection ],
|
||
|
|
function( sect )
|
||
|
|
return LoopByCayleyTable( Set ( CayleyTableByPerms( sect ) ) );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O QuasigroupByRightSection( sect )
|
||
|
|
##
|
||
|
|
## Returns the quasigroup whose right section is the list of permutations
|
||
|
|
## <sect>.
|
||
|
|
|
||
|
|
InstallMethod( QuasigroupByRightSection,
|
||
|
|
"for a set of left translation maps",
|
||
|
|
[ IsPermCollection ],
|
||
|
|
function( sect )
|
||
|
|
return QuasigroupByCayleyTable( TransposedMat ( CayleyTableByPerms( sect ) ) );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O LoopByRightSection( sect )
|
||
|
|
##
|
||
|
|
## Returns the loop whose right section is the list of permutations <sect>.
|
||
|
|
## Since the order of translations in <sect> is determined by their
|
||
|
|
## image of the neutral element 1, we disregard the order.
|
||
|
|
|
||
|
|
InstallMethod( LoopByRightSection,
|
||
|
|
"for a set of left translation maps",
|
||
|
|
[ IsPermCollection ],
|
||
|
|
function( sect )
|
||
|
|
return LoopByCayleyTable( TransposedMat ( Set ( CayleyTableByPerms( sect ) ) ) );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O LOOPS_CayleyTableByRightFolder( G, H, T )
|
||
|
|
##
|
||
|
|
## Auxiliary operation.
|
||
|
|
##
|
||
|
|
## A right folder is a triple (G,H,T) such that G is a group, H is
|
||
|
|
## a subgroup of G, and T is a right transversal to H in G.
|
||
|
|
##
|
||
|
|
## Returns the multiplication table on {Hx: x in G} by Ht*Hs = H(ts).
|
||
|
|
##
|
||
|
|
## The multiplication table is a quasigroup if and only if
|
||
|
|
## T is a right transversal to every conjugate H^g in G.
|
||
|
|
|
||
|
|
InstallGlobalFunction( LOOPS_CayleyTableByRightFolder,
|
||
|
|
function( G, H, T )
|
||
|
|
local act, nT, actT, i, p, ct;
|
||
|
|
# act = action of G on right cosest G/H
|
||
|
|
act := ActionHomomorphism( G, RightCosets( G, H ), OnRight );
|
||
|
|
nT := Length( T );
|
||
|
|
# actT = permutations on G/H induced by elements of T
|
||
|
|
actT := [1..nT];
|
||
|
|
for i in [1..nT] do
|
||
|
|
actT[ i ] := T[ i ]^act;
|
||
|
|
od;
|
||
|
|
# the order of right cosets determined by T might not agree with the default order of right cosets ...
|
||
|
|
p := PermList( List( [1..nT], i -> 1^actT[ i ] ) );
|
||
|
|
ct := List( [1..nT], i -> ListPerm( p * actT[ i ] * p^(-1) ) );
|
||
|
|
for i in [1..nT] do
|
||
|
|
if ct[ i ] = [] then # this can happen since ListPerm( () ) = []
|
||
|
|
ct[ i ] := [1..nT];
|
||
|
|
fi;
|
||
|
|
od;
|
||
|
|
return TransposedMat( ct );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O QuasigroupByRightFolder( G, H, T )
|
||
|
|
##
|
||
|
|
## See CayleyTableByRightFolder. We do not check if the right folder
|
||
|
|
## is a quasigroup right folder.
|
||
|
|
|
||
|
|
InstallMethod( QuasigroupByRightFolder,
|
||
|
|
"for a group, a subgroup and right transversal",
|
||
|
|
[ IsGroup, IsGroup, IsMultiplicativeElementCollection ],
|
||
|
|
function( G, H, T )
|
||
|
|
return QuasigroupByCayleyTable( LOOPS_CayleyTableByRightFolder( G, H, T ) );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O LoopByRightFolder( G, H, T )
|
||
|
|
##
|
||
|
|
## See CayleyTableByRigthFolder. We do not check if the right folder
|
||
|
|
## is a loop right folder.
|
||
|
|
|
||
|
|
InstallOtherMethod( LoopByRightFolder,
|
||
|
|
"for a group, a subgroup and right transversal",
|
||
|
|
[ IsGroup, IsGroup, IsMultiplicativeElementCollection ],
|
||
|
|
function( G, H, T )
|
||
|
|
return LoopByCayleyTable( LOOPS_CayleyTableByRightFolder( G, H, T ) );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
## CONVERSIONS
|
||
|
|
## -------------------------------------------------------------------------
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O IntoQuasigroup( M )
|
||
|
|
##
|
||
|
|
## Given a magma, returns the corresponding quasigroup, if possible.
|
||
|
|
|
||
|
|
InstallMethod( IntoQuasigroup, "for magma",
|
||
|
|
[ IsMagma ],
|
||
|
|
function( M )
|
||
|
|
local ct;
|
||
|
|
if IsQuasigroup( M ) then # leave quasigroups and loops intact
|
||
|
|
return M;
|
||
|
|
fi;
|
||
|
|
# magma, not necessarily a quasigroup
|
||
|
|
ct := MultiplicationTable( Elements( M ) );
|
||
|
|
if IsQuasigroupTable( ct ) then
|
||
|
|
return QuasigroupByCayleyTable( ct );
|
||
|
|
fi;
|
||
|
|
return fail;
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O PrincipalLoopIsotope( Q, f, g )
|
||
|
|
##
|
||
|
|
## Let Q be a quasigroup and f, g elements of Q.
|
||
|
|
## Define new operation on Q by x+y = R^{-1}(g)(x) * L^{-1}(f)(y).
|
||
|
|
## Then (Q,+) is a loop with neutral element f*g.
|
||
|
|
## We return isomorphic copy of (Q,+) via the isomorphism (1,f*g).
|
||
|
|
|
||
|
|
InstallMethod( PrincipalLoopIsotope,
|
||
|
|
"for quasigroup and two quasigroup elements",
|
||
|
|
[ IsQuasigroup, IsQuasigroupElement, IsQuasigroupElement ],
|
||
|
|
function( Q, f, g )
|
||
|
|
local n, L, R, i, j, ct, p;
|
||
|
|
if not (f in Q and g in Q) then
|
||
|
|
Error("LOOPS: <2> and <3> must be elements of quasigroup <1>.");
|
||
|
|
fi;
|
||
|
|
# constructing new multiplication
|
||
|
|
n := Size( Q );
|
||
|
|
ct := List( [1..n], i -> [1..n] );
|
||
|
|
L := Inverse( LeftTranslation( Q, f ) ); # inverse of left translation by f
|
||
|
|
R := Inverse( RightTranslation( Q, g ) ); # inverse or right translation by g
|
||
|
|
for i in [1..n] do for j in [1..n] do
|
||
|
|
ct[ i ][ j ] := CayleyTable( Q )[ i^R ][ j^L ];
|
||
|
|
od; od;
|
||
|
|
# the neutral element of ct is now f*g. We apply isomorphism (1, f*g).
|
||
|
|
p := Position(Q, f*g);
|
||
|
|
if p>1 then
|
||
|
|
p := (1, p); # note that p is its own inverse
|
||
|
|
ct := List([1..n], i-> List([1..n], j -> ( ct[ i^p ][ j^p ] )^p ) );
|
||
|
|
fi;
|
||
|
|
return LoopByCayleyTable( ct );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O IntoLoop( M )
|
||
|
|
##
|
||
|
|
## Given a magma, returns the corresponding loop, if possible.
|
||
|
|
|
||
|
|
InstallMethod( IntoLoop, "for magma",
|
||
|
|
[ IsMagma ],
|
||
|
|
function( M )
|
||
|
|
local e, p, ct;
|
||
|
|
if IsLoop( M ) then # loops are left intact
|
||
|
|
return M;
|
||
|
|
fi;
|
||
|
|
# magma, not necessarily a loop
|
||
|
|
M := IntoQuasigroup( M );
|
||
|
|
if M = fail then
|
||
|
|
return fail;
|
||
|
|
fi;
|
||
|
|
# quasigroup, not necesarily a loop
|
||
|
|
e := MultiplicativeNeutralElement( M );
|
||
|
|
if e = fail then # no neutral element, use principal isotope
|
||
|
|
return PrincipalLoopIsotope( M, Elements( M )[ 1 ], Elements( M )[ 1 ] );
|
||
|
|
fi;
|
||
|
|
# quasigroup with neutral element, i.e., a loop
|
||
|
|
p := Position( M, e );
|
||
|
|
if p>1 then
|
||
|
|
p := (1,p); # note that p is its own inverse
|
||
|
|
ct := List([1..Size(M)], i-> List([1..Size(M)], j ->
|
||
|
|
( CayleyTable( M )[ i^p ][ j^p ] )^p
|
||
|
|
) );
|
||
|
|
else
|
||
|
|
ct := CayleyTable( M );
|
||
|
|
fi;
|
||
|
|
return LoopByCayleyTable( ct );
|
||
|
|
end );
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O IntoGroup( M )
|
||
|
|
##
|
||
|
|
## Given a magma <M>, returns the corresponding group, if possible.
|
||
|
|
|
||
|
|
InstallOtherMethod( IntoGroup, "for magma",
|
||
|
|
[ IsMagma ],
|
||
|
|
function( M )
|
||
|
|
if IsGroup( M ) then # groups are left intact
|
||
|
|
return M;
|
||
|
|
fi;
|
||
|
|
# magma, not necessarily a group
|
||
|
|
M := IntoLoop( M );
|
||
|
|
if M=fail or (not IsAssociative( M ) ) then
|
||
|
|
return fail;
|
||
|
|
fi;
|
||
|
|
# group
|
||
|
|
return RightMultiplicationGroup( M );
|
||
|
|
end);
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
## PRODUCTS OF QUASIGROUPS AND LOOPS
|
||
|
|
## -------------------------------------------------------------------------
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#F DirectProduct( Q1, Q2, ..., Qn )
|
||
|
|
##
|
||
|
|
## Returns the direct product of quasigroups <Q1>, <Q2>, ... , <Qn>.
|
||
|
|
## The quasigroups can be declared as quasigroups, loops or groups.
|
||
|
|
|
||
|
|
# The following is necessary due to implementation of DirectProduct for
|
||
|
|
# groups in GAP. The idea is as follows:
|
||
|
|
# We want to calculate direct product of quasigroups, loops and groups.
|
||
|
|
# If only groups are on the list, standard GAP DirectProduct will take care
|
||
|
|
# of it. If there are also some quasigroups or loops on the list,
|
||
|
|
# we must take care of it.
|
||
|
|
# However, we do not know if such a list will be processed with
|
||
|
|
# DirectProductOp( <IsList>, <IsGroup> ), or
|
||
|
|
# DirectProductOp( <IsList>, <IsQuasigroup> ),
|
||
|
|
# since this depends on which algebra is listed first.
|
||
|
|
# We therefore take care of both situations.
|
||
|
|
|
||
|
|
InstallOtherMethod( DirectProductOp, "for DirectProduct( <IsList>, <IsGroup> )",
|
||
|
|
[ IsList, IsGroup],
|
||
|
|
function( list, first )
|
||
|
|
local L, p;
|
||
|
|
|
||
|
|
# Check the arguments.
|
||
|
|
if IsEmpty( list ) then Error( "LOOPS: <1> must be nonempty." ); fi;
|
||
|
|
if not ForAny( list, IsQuasigroup ) then
|
||
|
|
# there are no quasigroups or loops on the list
|
||
|
|
TryNextMethod();
|
||
|
|
fi;
|
||
|
|
if ForAny( list, G -> (not IsGroup( G )) and (not IsQuasigroup( G ) ) ) then
|
||
|
|
# there are other objects beside groups, loops and quasigroups on the list
|
||
|
|
TryNextMethod();
|
||
|
|
fi;
|
||
|
|
|
||
|
|
# all arguments are groups, quasigroups or loops, and there is at least one loop
|
||
|
|
# making sure that a loop is listed first so that this method is not called again
|
||
|
|
for L in list do
|
||
|
|
if not IsGroup( L ) then
|
||
|
|
p := Position( list, L );
|
||
|
|
list[ 1 ] := L;
|
||
|
|
list[ p ] := first;
|
||
|
|
break;
|
||
|
|
fi;
|
||
|
|
od;
|
||
|
|
|
||
|
|
return DirectProductOp( list, list[ 1 ] );
|
||
|
|
end);
|
||
|
|
|
||
|
|
InstallOtherMethod( DirectProductOp, "for DirectProduct( <IsList>, <IsQuasigroup> )",
|
||
|
|
[ IsList, IsQuasigroup ],
|
||
|
|
function( list, dummy )
|
||
|
|
|
||
|
|
local group_list, quasigroup_list, group_product, are_all_loops,
|
||
|
|
n, i, nL, nM, TL, TM, T, j, k, s;
|
||
|
|
|
||
|
|
# check the arguments
|
||
|
|
if IsEmpty( list ) then
|
||
|
|
Error( "LOOPS: <1> must be nonempty." );
|
||
|
|
elif ForAny( list, G -> (not IsGroup( G )) and (not IsQuasigroup( G ) ) ) then
|
||
|
|
TryNextMethod();
|
||
|
|
fi;
|
||
|
|
|
||
|
|
# only groups, quasigroups and loops are on the list, with at least one non-group
|
||
|
|
group_list := Filtered( list, G -> IsGroup( G ) );
|
||
|
|
quasigroup_list := Filtered( list, G -> IsQuasigroup( G ) );
|
||
|
|
if not IsEmpty( group_list ) then # some groups are on the list
|
||
|
|
group_product := DirectProductOp( group_list, group_list[ 1 ] );
|
||
|
|
Add( quasigroup_list, IntoLoop( group_product ) );
|
||
|
|
fi;
|
||
|
|
# keeping track of whether all algebras are in fact loops
|
||
|
|
are_all_loops := ForAll( quasigroup_list, IsLoop );
|
||
|
|
|
||
|
|
# now only quasigroups and loops are on the list
|
||
|
|
n := Length( quasigroup_list );
|
||
|
|
if n=1 then
|
||
|
|
return quasigroup_list[ 1 ];
|
||
|
|
fi;
|
||
|
|
# at least 2 quasigroups and loops; we will not use recursion
|
||
|
|
# making all Cayley tables cannonical
|
||
|
|
for s in [1..n] do
|
||
|
|
quasigroup_list[ s ] := QuasigroupByCayleyTable( CanonicalCayleyTable( CayleyTable( quasigroup_list[ s ] ) ) );
|
||
|
|
od;
|
||
|
|
for s in [2..n] do
|
||
|
|
nL := Size( quasigroup_list[ 1 ] );
|
||
|
|
nM := Size( quasigroup_list[ s ] );
|
||
|
|
TL := CayleyTable( quasigroup_list[ 1 ] );
|
||
|
|
TM := CayleyTable( quasigroup_list[ s ] );
|
||
|
|
T := List( [1..nL*nM], j->[] );
|
||
|
|
|
||
|
|
# not efficient, but it does the job
|
||
|
|
for i in [1..nM] do for j in [1..nM] do for k in [1..nL] do
|
||
|
|
Append( T[ (i-1)*nL + k ], TL[ k ] + nL*(TM[i][j]-1) );
|
||
|
|
od; od; od;
|
||
|
|
quasigroup_list[ 1 ] := QuasigroupByCayleyTable( T );
|
||
|
|
od;
|
||
|
|
if are_all_loops then
|
||
|
|
return IntoLoop( quasigroup_list[1] );
|
||
|
|
fi;
|
||
|
|
return quasigroup_list[ 1 ];
|
||
|
|
end );
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
## OPPOSITE QUASIGROUPS AND LOOPS
|
||
|
|
## --------------------------------------------------------------------------
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O OppositeQuasigroup( Q )
|
||
|
|
##
|
||
|
|
## Returns the quasigroup opposite to the quasigroup <Q>.
|
||
|
|
|
||
|
|
InstallMethod( OppositeQuasigroup, "for quasigroup",
|
||
|
|
[ IsQuasigroup ],
|
||
|
|
function( Q )
|
||
|
|
return QuasigroupByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
|
||
|
|
end );
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#O OppositeLoop( Q )
|
||
|
|
##
|
||
|
|
## Returns the loop opposite to the loop <Q>.
|
||
|
|
|
||
|
|
InstallMethod( OppositeLoop, "for loop",
|
||
|
|
[ IsLoop ],
|
||
|
|
function( Q )
|
||
|
|
return LoopByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
|
||
|
|
end );
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
##
|
||
|
|
#A Opposite( Q )
|
||
|
|
##
|
||
|
|
## Returns the quasigroup opposite to the quasigroup <Q>. When
|
||
|
|
## <Q> is a loop, a loop is returned.
|
||
|
|
|
||
|
|
InstallMethod( Opposite, "for quasigroup",
|
||
|
|
[ IsQuasigroup ],
|
||
|
|
function( Q )
|
||
|
|
if IsLoop( Q ) then
|
||
|
|
return LoopByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
|
||
|
|
fi;
|
||
|
|
return QuasigroupByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
|
||
|
|
end );
|
||
|
|
|
||
|
|
#############################################################################
|
||
|
|
## DISPLAYING QUASIGROUPS AND LOOPS
|
||
|
|
## -------------------------------------------------------------------------
|
||
|
|
|
||
|
|
InstallMethod( ViewObj, "for quasigroup",
|
||
|
|
[ IsQuasigroup ],
|
||
|
|
function( Q )
|
||
|
|
Print( "<quasigroup of order ", Size( Q ), ">" );
|
||
|
|
end );
|
||
|
|
|
||
|
|
## dangerous for large quasigroups
|
||
|
|
InstallMethod( PrintObj, "for quasigroup",
|
||
|
|
[ IsQuasigroup ],
|
||
|
|
function( Q )
|
||
|
|
if HasCayleyTable( Q ) then
|
||
|
|
Print( "<quasigroup with multiplication table\n" );
|
||
|
|
PrintArray( CayleyTable( Q ) );
|
||
|
|
else
|
||
|
|
Print( "<quasigroup with elements\n" );
|
||
|
|
Print( Elements( Q ) );
|
||
|
|
fi;
|
||
|
|
Print( ">\n" );
|
||
|
|
end );
|
||
|
|
|
||
|
|
InstallMethod( ViewObj, "for loop",
|
||
|
|
[ IsLoop ],
|
||
|
|
function( L )
|
||
|
|
if HasIsAssociative( L ) and IsAssociative( L ) then
|
||
|
|
Print( "<associative loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then
|
||
|
|
Print( "<extra loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then
|
||
|
|
Print( "<Moufang loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsCLoop( L ) and IsCLoop( L ) then
|
||
|
|
Print( "<C loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then
|
||
|
|
Print( "<left Bol loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then
|
||
|
|
Print( "<right Bol loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsLCLoop( L ) and IsLCLoop( L ) then
|
||
|
|
Print( "<LC loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsRCLoop( L ) and IsRCLoop( L ) then
|
||
|
|
Print( "<RC loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsLeftAlternative( L ) and IsLeftAlternative( L ) then
|
||
|
|
if HasIsRightAlternative( L ) and IsRightAlternative( L ) then
|
||
|
|
Print( "<alternative loop of order ", Size( L ), ">");
|
||
|
|
else
|
||
|
|
Print( "<left alternative loop of order ", Size( L ), ">");
|
||
|
|
fi;
|
||
|
|
elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then
|
||
|
|
Print( "<right alternative loop of order ", Size( L ), ">");
|
||
|
|
elif HasIsFlexible( L ) and IsFlexible( L ) then
|
||
|
|
Print( "<flexible loop of order ", Size( L ), ">");
|
||
|
|
else
|
||
|
|
# MORE ??
|
||
|
|
Print( "<loop of order ", Size( L ), ">" );
|
||
|
|
fi;
|
||
|
|
end );
|
||
|
|
|
||
|
|
## dangerous for large loops
|
||
|
|
InstallMethod( PrintObj, "for loop",
|
||
|
|
[ IsLoop ],
|
||
|
|
function( L )
|
||
|
|
if HasCayleyTable( L ) then
|
||
|
|
Print( "<loop with multiplication table\n" );
|
||
|
|
PrintArray( CayleyTable( L ) );
|
||
|
|
else
|
||
|
|
Print( "<loop with elements\n" );
|
||
|
|
Print( Elements( L ) );
|
||
|
|
fi;
|
||
|
|
Print( ">\n" );
|
||
|
|
end );
|