  This is the second in a series of papers developing a theory of total
positivity for loop groups. In this paper, we study infinite products of
Chevalley generators. We show that the combinatorics of infinite reduced words
underlies the theory, and develop the formalism of infinite sequences of braid
moves, called a braid limit. We relate this to a partial order, called the
limit weak order, on infinite reduced words.
  The limit semigroup generated by Chevalley generators has a transfinite
structure. We prove a form of unique factorization for its elements, in effect
reducing their study to infinite products which have the order structure of the
natural numbers. For the latter infinite products, we show that one always has
a factorization which matches an infinite Coxeter element.
  One of the technical tools we employ is a totally positive exchange lemma
which appears to be of independent interest. This result states that the
exchange lemma (in the context of Coxeter groups) is compatible with total
positivity in the form of certain inequalities.
