{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:47:25Z","timestamp":1760143645651,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,24]],"date-time":"2024-02-24T00:00:00Z","timestamp":1708732800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations.<\/jats:p>","DOI":"10.3390\/axioms13030148","type":"journal-article","created":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T03:34:04Z","timestamp":1708918444000},"page":"148","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Continuous Multi-Utility Representations of Preorders and the Chipman Approach"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3799-4630","authenticated-orcid":false,"given":"Gianni","family":"Bosi","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Universit\u00e0 di Trieste, 34127 Trieste, Italy"}]},{"given":"Roberto","family":"Daris","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Universit\u00e0 di Trieste, 34127 Trieste, Italy"}]},{"given":"Magal\u00ec","family":"Zuanon","sequence":"additional","affiliation":[{"name":"Dipartimento di Economia e Management, DEM, Universit\u00e0 di Brescia, 25122 Brescia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1016\/j.jmateco.2003.06.002","article-title":"The Debreu Gap Lemma and some generalizations","volume":"40","author":"Herden","year":"2004","journal-title":"J. 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