I don't have the gang of four here at my cubicle, but I do develop code which uses decorators.

Decorator B should not "inherit" anything from A; they are more of a 'friend' relationship. In Perl, you would probably end up munging the symbol table to accomplish it. Decorators are added after-the-fact, per-instance, upon demand. So SOME instances of A also have B-like qualities, and SOME instances of A do not.

Another way of saying it: You don't really say "make a new instance of a B," but instead, "make a new instance of an A, and then attach some B-ness to it."

my $a = new A();
$a->bee();          # A::bee() does not exist or has no effect
$a->attach('B');    # make A load B and attach B methods
$a->bee();          # A::bee() now exists thanks to B::bee
[download]

This approach would still hold true, even if you make another Decorator like C. Some A instances may have B, some have C, some have both B and C, and some have neither.

Update: You can then make a D class which inherits from A, but always attaches B-ness and C-ness.