I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the well-founded part of a model? If someone could give me a precise definition (maybe it can be defined using transitive closures, but I don't really know) of the well-founded part of a model, it'd be greatly appreciated.
Addendum
The well-foundedness that I'm referring to is not the internal well-foundedness that comes from assuming the Axiom of Regularity within the model. It's an external property, as viewed from outside the model.
Suppose that $(M,E)$ is a model of ZFC, this is a set in the universe (which is also a model of ZFC, for our purposes).
It is possible that $(M,E)$ is not a well-founded relation. Internally, of course, this is impossible. $M$ does not have any element which is a decreasing sequence in $E$, since $M$ satisfies the axiom of regularity.
However we, as educated men staring at $M$ externally, know that it is possible that $M$ has more than it knows about. One can now ask about the ordinals of $M$. Namely $(Ord^M,E)$ as a linear order. This order has a maximal initial segment which is well-founded.
The well-founded part is the initial part [internally] of $(M,E)$ which is truly well-founded. It is exactly the sets whose [internal] von Neumann rank is an ordinal in the well-founded part of $(Ord^M,E)$.