2.24 Monoids of Relations

There are no special functions provided for monoids generated by binary relations. The action of such a monoid on sets, however, provides a way to convert a relation monoid into a transformation monoid (see chapter Actions of Monoids). This monoid can then be used to investigate the structure of the original relation monoid.

```    gap> a:= Relation( [ [  ], [  ], [ 1, 3, 4 ], [  ], [ 2, 5 ] ] );;
gap> b:= Relation( [ [  ], [ 2 ], [ 4 ], [ 1, 2, 3 ], [ 1 ] ] );;
gap> M:= Monoid( a, b );
Monoid( [ Relation( [ [  ], [  ], [ 1, 3, 4 ], [  ], [ 2, 5 ] ] ),
Relation( [ [  ], [ 2 ], [ 4 ], [ 1, 2, 3 ], [ 1 ] ] ) ] )
gap> # transform points into singleton sets.
gap> one:= List( [ 1 .. 5 ], x-> [ x ] );
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ] ]
gap> # determine all reachable sets.
gap> sets:= Union( Orbits( M, one ) );
[ [  ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 2, 3, 4 ], [ 1, 3, 4 ],
[ 2 ], [ 2, 4 ], [ 2, 5 ], [ 3 ], [ 4 ], [ 5 ] ]
gap> # construct isomorphic transformation monoid.
gap> act:= Action( M, sets );
Monoid( [ Transformation( [ 1, 1, 1, 6, 6, 6, 1, 1, 9, 6, 1, 9 ] ),
Transformation( [ 1, 1, 7, 8, 5, 5, 7, 4, 3, 11, 4, 2 ] ) ] )
gap> Size(act);
11```

Version 2.4 (May 1998)