I'm basically only interested in the above question. I have played with this a little bit and have not found an obvious way to prove it. However, I am not familiar with ways for proving something is semi-artinian in general (i.e. what seem to be set-theoretic methods, or discussions of the so-called socles).
If this is not generally true, is it likely to be true for the power set of the natural numbers?
Thanks for any help on this.