On a geometrical notion of dimension for partially ordered sets
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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 new notion of dimension for partial orders called Debreu dimension, a variation of the Dushnik-Miller dimension that is closer to geometry. Instead of arbitrary linear extensions as considered by the Dushnik-Miller dimension, we consider specifically Debreu separable linear extensions. Our main result is that the Debreu dimension is countable if and only if countable multi-utilities exist. Importantly, unlike the classical results concerning linear extensions like the Szpilrajn extension theorem, we avoid using the axiom of choice. Instead, we rely on countability restrictions to sequentially construct extensions which, in the limit, yield Debreu separable linear extensions.
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