  The irreducible euclidean Coxeter groups that naturally act geometrically on
euclidean space are classified by the well-known extended Dynkin diagrams and
these diagrams also encode the modified presentations that define the
irreducible euclidean Artin groups. These Artin groups have remained mysterious
with some exceptions until very recently. Craig Squier clarified the structure
of the three examples with three generators more than twenty years ago and
Fran\c{c}ois Digne more recently proved that two of the infinite families can
be understood by constructing a dual presentation for each of these groups and
showing that it forms an infinite-type Garside structure. In this article I
establish that none of the remaining dual presentations for irreducible
euclidean Artin groups corrspond to Garside structures because their
factorization posets fail to be lattices. These are the first known examples of
dual Artin presentations that fail to form Garside structures. Moreover, the
results presented here about the cause of this failure form the foundation for
a subsequent article in which the structure of all euclidean Artin groups is
clarified.
