TITLE: What is the definition of the well-founded part of a model of set theory?
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: This is a definition taken from the proof of Theorem 47 of Azriel Lévy’s monograph, A Hierarchy of Formulas in Set Theory (Memoirs of the AMS, Number 57).

Definition. Let $ M $ be a set, and $ E $ a binary relation on $ M $.

A subset $ X $ of $ M $ is called $ E $-transitive if and only if
  $$
(\forall x,y \in M)(((y \in X) \land ((x,y) \in E)) \to (x \in X)).
$$
The $ E $-transitive closure of an element $ x $ of $ M $ is defined as the
  following subset of $ M $:
  $$
\{
y \in M \mid
(\forall X)
(
((x \in X \subseteq M) \land (X ~ \text{is} ~ E \text{-transitive})) \to
(y \in X)
)
\}.
$$
A subset $ X $ of $ M $ is called $ E $-well-founded if and only if for
  every non-empty subset $ Y $ of $ X $, there exists a $ y \in Y $ such that
  $ (x,y) \notin E $ for every $ x \in Y \setminus \{ y \} $.
The $ E $-well-founded part of $ M $ is finally defined as the following
  subset of $ M $:
  $$
\{
x \in M \mid
\text{The} ~ E \text{-transitive closure of} ~ x ~ \text{is} ~ E
\text{-well-founded}
\}.
$$