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. This notion is of particular importance in economics and physics, since it constitutes the system's degrees of freedom. Here, we introduce a new notion of dimension for partial orders called Debreu dimension, a variation of the Dushnik-Miller dimension that is closer to geometry and is based on Debreu separable linear extensions. Our main results are the following: (i) under some countability restrictions, Debreu separable linear extensions can be obtained as the limit of a sequence of partial orders that extend the original one and, moreover, linear extensions can be constructed in a similar fashion from monotones, and (ii) the Debreu dimension is countable if and only if countable multi-utilities exist, although there are partial orders with finite multi-utilities where the Debreu dimension is countably infinite. As an application of (ii), we improve on the classification of preorders through real-valued monotones by showing that there are preorders where finite multi-utilities exist and finite strict monotone multi-utilities do not.