🤖 AI Summary
A fundamental information asymmetry exists between measurement-based mathematical representations and optimization-principle-based physical representations of experimental setups: certain partial orders require infinite information for physical realization, yet admit finite-information mathematical descriptions. Method: We introduce the “Debreu dimension”—a novel dimension concept unifying geometric intuition with order-theoretic rigor—thereby bridging the semantic gap between Dushnik–Miller dimension and classical geometric dimension. Under countability assumptions, we develop an explicit, geometry-guided constructive framework, integrating order theory, real-valued monotone function analysis, and set-theoretic tools. Contribution/Results: This work establishes a comprehensive dimensional classification scheme for preordered spaces, significantly enhancing both the geometric unification of partially ordered sets and the classification accuracy of real-valued monotone representations.
📝 Abstract
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.
