Vtyjkyuk

How would you describe a function that has the relationship $fleft(fleft(fleft(phi, xright), yright), zright) == fleft(phi, zright)$? given that $fleft(phi, xright)$ is idempotent (i.e., $fleft(fleft(phi, xright), xright) == fleft(phi, xright)$?

Is this simply a property of idempotent functions?

An application of this function can be seen in a programming language where a method object.SetProperty(x) changes the state of the object. However, subsequent calls to SetProperty() overwrites the existing state. Thus the following are considered equivalent:

object = new Object;
object.SetProperty(x);
object.SetProperty(y);
object.SetProperty(z);

and

object = new Object;
object.SetProperty(z);

I think the property you are looking for is the following:
$$
f(f(phi, x),y) = f(phi,y) text for all x, y. tag1
$$
If $f$ satisfies (1), then this both implies that $f$ is idempotent (by your definition) and also implies your condition, $fleft(fleft(fleft(phi, xright), yright), zright) = fleft(phi, zright)$.

So, I think (1) is the condition you want. I am not aware of a name for this property, except that it is similar to something called a "right zero" or "right-absorbing-element". So, I would suggest that if $f$ satisfies (1), then you could call $f$ right-absorbing. This is meant to refer to the fact that the right-most applied argument $y$ "absorbs" the other applied argument $x$.

Is this simply a property of idempotent functions?

No, the property (1) is not true of all idempotent functions. For example, if $phi$ is a set and $f(phi, x)$ represents adding the element $x$ to $phi$ (phi.add(x)), this is idempotent (because adding the same element twice to a set is the same as adding it once), but not right-absorbing (because adding $x$ and then adding $y$ isn't the same as just adding $y$).

I have never heard of a particular name for that property in that context, however in probability there is the adjective memoryless which corresponds to this idea.

See Poisson / Exponential variables, as well as Markov processes.