[Haskell-cafe] Is there a name for this algebraic structure?

Hello I am wondering if there's a well known algebraic structure which follows the following patterns. Let's call it S: It's update-able with some opaque "a" (which can be an element or an operation with an element): update :: S -> a -> S There's a well defined zero for it: empty :: S Operations on it are idempotent: update s a == update (update s a) a Every S can be reconstructed from a sequence of updates: forall s. exists [a]. s == foldl update empty [a] An example of this would be Data.Set: empty = Set.empty update = flip Set.insert Is there something like this in algebra? Cheers, Gleb