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

[Gleb Peregud]

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
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