  Given a totally finite ordered alphabet $ A $, endowing the set of words over
$ A $ with the alternating lexicographic order, we define a new class of Lyndon
words. We study the fundamental properties of the associated symbolic dynamical
systems called Lyndon system. We derive some fundamental properties of the
beta-shift with negative base by relating it with the Lyndon system. We find,
independently of W. Steiner's method, the conditions for which a word is the
$(-\beta)$-expansion of $-\frac{\beta}{\beta + 1} $ for some $ \beta> 1$.
