  In this paper, we continue the study of a left-distributive algebra of
elementary embeddings from the collection of sets of rank less than lambda to
itself, as well as related finite left-distributive algebras (which can be
defined without reference to large cardinals). In particular, we look at the
critical points (least ordinals moved) of the elementary embeddings; simple
statements about these ordinals can be reformulated as purely algebraic
statements concerning the left distributive law. Previously, lower bounds on
the number of critical points have been used to show that certain such
algebraic statements, known to follow from large cardinals, require more than
Primitive Recursive Arithmetic to prove. Here we present the first few steps of
a program that, if it can be carried to completion, should give exact
computations of the number of critical points, thereby showing that hypotheses
only slightly beyond Primitive Recursive Arithmetic would suffice to prove the
aforementioned algebraic statements.
  Some of the main results here were discovered through examination of bit
patterns from extensive calculations in the finite algebras; this may be the
first serious example of computer-aided research in the theory of large
cardinals. The results in this paper have a large overlap with results of A.
Drapal, which were obtained independently at about the same time, but from a
quite different point of view.
