Lift some functions to magmas in general

gap/quasigroups.gd

@@ -96,11 +96,12 @@ DeclareOperation( "NormalizedQuasigroupTable", [ IsMatrix ] );
 ##  CREATING QUASIGROUPS AND LOOPS MANUALLY
 ##  -------------------------------------------------------------------------
 
-DeclareAttribute( "CayleyTable", IsQuasigroup );
+DeclareAttribute( "CayleyTable", IsMagma );
 DeclareOperation( "QuasigroupByCayleyTable", [ IsMatrix ] );
 DeclareOperation( "LoopByCayleyTable", [ IsMatrix ] );
 DeclareOperation( "SetQuasigroupElmName", [ IsQuasigroup, IsString ] );
 DeclareSynonym( "SetLoopElmName", SetQuasigroupElmName );
+DeclareAttribute( "ConstructorFromTable", IsMagma );
 DeclareOperation( "CanonicalCopy", [ IsMagma ] );
 
 #############################################################################
@@ -142,9 +143,9 @@ DeclareOperation( "IntoGroup", [ IsMagma ] );
 ## OPPOSITE QUASIGROUPS AND LOOPS
 ## --------------------------------------------------------------------------
 
-DeclareOperation( "OppositeQuasigroup", [ IsQuasigroup ] );
+DeclareGlobalFunction( "OppositeQuasigroup");
-DeclareOperation( "OppositeLoop", [ IsLoop ] );
+DeclareGlobalFunction( "OppositeLoop");
-DeclareAttribute( "Opposite", IsQuasigroup );
+DeclareAttribute( "Opposite", IsMagma );
 
 #############################################################################
 ## AUXILIARY

gap/quasigroups.gi

@@ -149,10 +149,12 @@ end );
 ##
 #A  CayleyTable( Q )
 ##
-##  Returns the Cayley table of the quasigroup <Q>
+##  Returns the Cayley table of the magma <Q>. This is just like
+##  its multiplication table, except in case Q is a submagma of P, in which
+##  case the entries used are the indices in P rather than in Q.
 
-InstallMethod( CayleyTable, "for quasigroup",
+InstallMethod( CayleyTable, "for magma",
-    [ IsQuasigroup ],
+    [ IsMagma ],
 function( Q )
     local elms, parent_elms;
     elms := Elements( Q );
@@ -194,6 +196,7 @@ function( ct )
     SetAsSSortedList( Q, elms );
     SetParent( Q, Q );
     SetCayleyTable( Q, ct );
+    SetConstructorFromTable(QuasigroupByCayleyTable);
     return Q;
 end );
 
@@ -232,6 +235,7 @@ function( ct )
     SetParent( L, L );
     SetCayleyTable( L, ct );
     SetOne( L, elms[ 1 ] );
+    SetConstructorFromTable(LoopByCayleyTable);
     return L;
 end );
 
@@ -249,22 +253,50 @@ function( Q, name )
     F!.names := name;
 end);
 
+#############################################################################
+##
+#O ConstructorFromTable( M )
+##
+## Given a magma <M>, returns a function which will create a domain of the
+## same structure as M from from an operation table. This implementation of
+## the method is to backfill the constructors for library domains that do
+## not set the attribute directly at construction time. New domains that wish
+## to use facilities like CanonicalCopy or Opposite should call
+## SetConstructorFromTable at creation time.
+
+InstallMethod( ConstructorFromTable, "for other magmas",
+    [ IsMagma ],
+    function ( M ) 
+      # Go in reverse order of refinement of structure
+      if IsGroup(M) then
+        return GroupByMultiplicationTable;
+      elif IsMagmaWithInverses(M) then
+        return MagmaWithInversesByMultiplicationTable;
+      elif IsMonoid(M) then
+        return MonoidByMultiplicationTable;
+      elif IsMagmaWithOne(M) then
+        return MagmaWithOneByMultiplicationTable;
+      elif IsSemigroup(M) then
+        return SemigroupByMultiplicationTable;
+      fi;
+      return MagmaByMultiplicationTable;
+end);
+
+            
 #############################################################################
 ##
 #O  CanonicalCopy( Q )
 ##
 ##  Returns a canonical copy of <Q>, that is, an isomorphic object <O> with
-##  canonical multiplication table and Parent( <O> ) = <O>.
+##  canonical multiplication table and no parent set. Note that this is
+##  guaranteed to be a new object, not satisfying IsIdenticalObj with any
+##  previously existing structure. THEREFORE:
 ##  (PROG) Properties and attributes are lost!
 
-InstallMethod( CanonicalCopy, "for quasigroup or loop",
+InstallMethod( CanonicalCopy, "for magma",
-    [ IsQuasigroup ],
+    [ IsMagma ],
-function( Q )
+    M -> ConstructorFromTable(M)(MultiplicationTable(M));
-    if IsLoop( Q ) then
+);
-        return LoopByCayleyTable( CayleyTable( Q ) );
-    fi;
-    return QuasigroupByCayleyTable( CayleyTable( Q ) );
-end);
 
 
 #############################################################################
@@ -768,41 +800,31 @@ function( list, dummy )
 ##
 #O  OppositeQuasigroup( Q )
 ##
-##  Returns the quasigroup opposite to the quasigroup <Q>. 
+##  Identical to Opposite, except forces its return to be a quasigroup if
+##  possible 
 
-InstallMethod( OppositeQuasigroup, "for quasigroup",
+InstallGlobalFunction( OppositeQuasigroup,
-    [ IsQuasigroup ],
+                       Q -> IntoQuasigroup( Opposite( L ) ) );
-function( Q )
-    return QuasigroupByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
-end );
 
 #############################################################################
 ##
 #O  OppositeLoop( Q )
 ##
-##  Returns the loop opposite to the loop <Q>. 
+##  Identical to Opposite, except forces its return to be a loop if possible
 
-InstallMethod( OppositeLoop, "for loop",
+InstallGlobalFunction( OppositeLoop, L -> IntoLoop( Opposite( L ) ) );
-    [ IsLoop ],
-function( Q )
-    return LoopByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
-end );
 
 #############################################################################
 ##
-#A  Opposite( Q )
+#A  Opposite( M )
 ##
-##  Returns the quasigroup opposite to the quasigroup <Q>. When
+##  Returns the magma opposite to the magma <M>, with as much structure
-##  <Q> is a loop, a loop is returned.
+##  as can be preserved.
 
-InstallMethod( Opposite, "for quasigroup",
+InstallMethod( Opposite, "for magma",
-    [ IsQuasigroup ],
+    [ IsMagma ],
-function( Q )
+    M -> ConstructorFromTable(M)( TransposedMat( MultiplicationTable( M ) ) )
-    if IsLoop( Q ) then
+);
-        return LoopByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
-    fi;
-    return QuasigroupByCayleyTable( TransposedMat( CayleyTable( Q ) ) );
-end );
 
 #############################################################################
 ##  DISPLAYING QUASIGROUPS AND LOOPS