  We will study the presentations of fundamental groups of the complement of
complexified real affine line arrangements that do not contain two parallel
lines. By Yoshinaga's minimal presentation, we can give positive homogeneous
presentations of the fundamental groups. We consider the associated monoids
defined by the presentations. It turns out that, in some cases, left (resp.
right) \emph{least common multiple} does not always exist. Hence, the monoids
are neither \emph{Garside} nor \emph{Artin}. Nevertheless, we will show that
they carry certain particular elements similar to the \emph{fundamental
elements} in Artin monoids, and that, by improving the classical method in
combinatorial group theory, they are \emph{cancellative monoids}. As a result,
we will show that the word problem can be solved and the center of them are
determined.
