On a geometrical notion of dimension for partially ordered sets

The well-known notion of dimension for partial orders by Dushnik and Miller allows to quantify the degree of incomparability and, thus, is regarded as a measure of complexity for partial orders. However, despite its usefulness, its definition is somewhat disconnected from the geometrical idea of dimension, where, essentially, the number of dimensions indicates how many real lines are required to represent the underlying partially ordered set. Here, we introduce a variation of the Dushnik-Miller notion of dimension that is closer to geometry, the Debreu dimension, and show the following main results: (i) how to construct its building blocks under some countability restrictions, (ii) its relation to other notions of dimension in the literature, and (iii), as an application of the above, we improve on the classification of preordered spaces through real-valued monotones.