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On the threshold representation of partial orderings

Daniele Caliari & Davide Carpentiere

Mathematical Social Sciences2026https://doi.org/10.1016/j.mathsocsci.2026.102516article
AJG 2ABDC A
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Abstract

In this paper, we provide a constructive proof for threshold representations of partial orderings in a finite setting and extend the construction to arbitrary sets. This contributes to a long-standing open problem posed in Nakamura (2002) and Aleskerov et al. (2007). We identify and discuss aspects of our proof that do not directly generalize to an infinite setting and, for each, provide intuitive sufficient conditions. • ≻ is a partial order if and only if it is metric representable (finite case). • Constructive proof via the minimum weighted path metric. • The representation is extended to infinite sets under mild conditions. • These resolve an open problem posed in Nakamura (2002) and Aleskerov et al. (2007).

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https://doi.org/https://doi.org/10.1016/j.mathsocsci.2026.102516

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@article{daniele2026,
  title        = {{On the threshold representation of partial orderings}},
  author       = {Daniele Caliari & Davide Carpentiere},
  journal      = {Mathematical Social Sciences},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1016/j.mathsocsci.2026.102516},
}

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On the threshold representation of partial orderings

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0.50

Balanced mode · F 0.40 / M 0.15 / V 0.05 / R 0.40

F · citation impact0.50 × 0.4 = 0.20
M · momentum0.50 × 0.15 = 0.07
V · venue signal0.50 × 0.05 = 0.03
R · text relevance †0.50 × 0.4 = 0.20

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