Upper Semicontinuous Representations of Semiorders as Interval Orders

Bosi, Gianni; Sbaiz, Gabriele; Zuanon, Magalì

doi:10.3390/axioms15010053

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Upper Semicontinuous Representations of Semiorders as Interval Orders

by

Gianni Bosi

^1,*

,

Gabriele Sbaiz

¹

and

Magalì Zuanon

²

¹

Department of Economics, Business, Mathematics and Statistics, University of Trieste, Via Valerio 4/1, 34127 Trieste, Italy

²

Department of Economics and Management, University of Brescia, Contrada Santa Chiara 50, 25122 Brescia, Italy

^*

Author to whom correspondence should be addressed.

Axioms 2026, 15(1), 53; https://doi.org/10.3390/axioms15010053

Submission received: 11 December 2025 / Revised: 8 January 2026 / Accepted: 9 January 2026 / Published: 10 January 2026

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics, 2nd Edition)

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Abstract

We characterize the upper semicontinuous representability of a semiorder ≺ as an interval order (namely, by a pair

(u, v)

of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the representing functions is a one-way utility for the characteristic weak order

≺^{0}

associated with the semiorder. Such a description generalizes the upper semicontinuous threshold representation. To this end, we introduce a suitable upper semicontinuity condition concerning a semiorder, namely strict upper semicontinuity. We further characterize the mere existence of an upper semicontinuous one-way utility for this characteristic weak order, with a view to the identification of maximal elements on compact metric spaces.

Keywords: semiorder; upper semicontinuous representations by means of two functions; traces; one-way utility

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MDPI and ACS Style

Bosi, G.; Sbaiz, G.; Zuanon, M. Upper Semicontinuous Representations of Semiorders as Interval Orders. Axioms 2026, 15, 53. https://doi.org/10.3390/axioms15010053

AMA Style

Bosi G, Sbaiz G, Zuanon M. Upper Semicontinuous Representations of Semiorders as Interval Orders. Axioms. 2026; 15(1):53. https://doi.org/10.3390/axioms15010053

Chicago/Turabian Style

Bosi, Gianni, Gabriele Sbaiz, and Magalì Zuanon. 2026. "Upper Semicontinuous Representations of Semiorders as Interval Orders" Axioms 15, no. 1: 53. https://doi.org/10.3390/axioms15010053

APA Style

Bosi, G., Sbaiz, G., & Zuanon, M. (2026). Upper Semicontinuous Representations of Semiorders as Interval Orders. Axioms, 15(1), 53. https://doi.org/10.3390/axioms15010053

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Upper Semicontinuous Representations of Semiorders as Interval Orders

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