I was asking generally. I guess the question was "Does any of the systems you looked at so far have composition?" (I realize I should have specified better 😅 )
The peg based and gate (what I call an or gate because that makes more sense to me, though in the end it of course does not matter), kind of has the issue that you cannot compose them trivially as the output is in the form of a closed string while inputs (pegs) are open strings. So you need the group theory.