Let be the set of all strictly increasing sequences of integers. Define a binary operator by . The pair is a monoid. Define a sequence of functions for , by . It is easy enough to check that each function in the sequence is a homomorphism, and for each , . Let . Questions: Is a sub-monoid of ? Is there a way to "mod-out" by ?

Edit:

Some additional info:

The identity element is the sequence . The sets, are all closed under addition. Does the set ? If so, it should be closed under addition, as well. I forgot to mention that is commutative. So, if "modding out" does make sense in this context, cosets would have the form . Without inverses, I have no idea how to tell if two sequences are in the same left coset.