  A generalized lexicographical order on infinite words is defined by choosing
for each position a total order on the alphabet. This allows to define
generalized Lyndon words. Every word in the free monoid can be factorized in a
unique way as a nonincreasing factorization of generalized Lyndon words. We
give new characterizations of the first and the last factor in this
factorization as well as new characterization of generalized Lyndon words. We
also give more specific results on two special cases: the classical one and the
one arising from the alternating lexicographical order.
