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The irreducible euclidean Coxeter groups that naturally act geometrically oneuclidean space are classified by the well-known extended Dynkin diagrams andthese diagrams also encode the modified presentations that define theirreducible euclidean Artin groups. These Artin groups have remained mysteriouswith some exceptions until very recently. Craig Squier clarified the structureof the three examples with three generators more than twenty years ago andFran\c{c}ois Digne more recently proved that two of the infinite families canbe understood by constructing a dual presentation for each of these groups andshowing that it forms an infinite-type Garside structure. In this article Iestablish that none of the remaining dual presentations for irreducibleeuclidean Artin groups corrspond to Garside structures because theirfactorization posets fail to be lattices. These are the first known examples ofdual Artin presentations that fail to form Garside structures. Moreover, theresults presented here about the cause of this failure form the foundation fora subsequent article in which the structure of all euclidean Artin groups isclarified.

Dual euclidean Artin groups and the failure of the lattice property