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