{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:07Z","timestamp":1760147347728,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T00:00:00Z","timestamp":1674777600000},"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>We present some lifting theorems for continuous order-preserving functions on locally and \u03c3-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and \u03c3-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of \u227e-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions.<\/jats:p>","DOI":"10.3390\/axioms12020123","type":"journal-article","created":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T02:30:38Z","timestamp":1674786638000},"page":"123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3799-4630","authenticated-orcid":false,"given":"Gianni","family":"Bosi","sequence":"first","affiliation":[{"name":"Department of Economics, Business, Mathematics and Statistics, University of Trieste, Via A. Valerio 4\/1, 1, 34127 Trieste, Italy"}]},{"given":"Magal\u00ec","family":"Zuanon","sequence":"additional","affiliation":[{"name":"Dipartimento di Economia e Management, Universit\u00e0 di Brescia, 25122 Brescia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bosi, G. (2022). Continuous Order-Preserving Functions for All Kind of Preorders. Order.","DOI":"10.1007\/s11083-022-09598-3"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bosi, G., and Zuanon, M. (2022). Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions. Axioms, 11.","DOI":"10.3390\/axioms11050195"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1017\/S0305004100053688","article-title":"On separation axioms for certain types of ordered topological space","volume":"82","author":"Burgess","year":"1977","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gillman, L., and Jerison, M. (1960). Rings of Continuous Functions, D. Van Nostrand Company.","DOI":"10.1007\/978-1-4615-7819-2"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/0165-4896(89)90058-9","article-title":"On the existence of utility functions","volume":"17","author":"Herden","year":"1989","journal-title":"Math. Social Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/0165-4896(89)90041-3","article-title":"On the existence of utility functions II","volume":"18","author":"Herden","year":"1989","journal-title":"Math. Social Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0165-4896(01)00077-4","article-title":"On the continuous analogue of the Szpilrajn theorem","volume":"43","author":"Herden","year":"2002","journal-title":"Math. Soc. Sci."},{"key":"ref_8","first-page":"715","article-title":"A continuous utility theorem for closed preorders on a \u03c3\u2014Compact metrizable space","volume":"28","author":"Levin","year":"1983","journal-title":"Sov. Math. Dokl."},{"key":"ref_9","first-page":"639","article-title":"Measurable utility theorem for closed and lexicographic preference relations","volume":"27","author":"Levin","year":"1983","journal-title":"Sov. Math. Dokl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1017\/S0004972700004597","article-title":"Existence of an order-preserving function on a normally preordered space","volume":"34","author":"Mehta","year":"1986","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/0165-4896(88)90018-2","article-title":"Some general theorems on the existence of order-preserving functions","volume":"15","author":"Mehta","year":"1988","journal-title":"Math. Soc. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1016\/j.topol.2013.03.008","article-title":"Convexity and quasi-uniformizability of closed preordered spaces","volume":"160","author":"Minguzzi","year":"2013","journal-title":"Topol. Appl."},{"key":"ref_13","unstructured":"Nachbin, L. (1965). Topology and Order, Van Nostrand."},{"key":"ref_14","unstructured":"Engelking, R. (1989). General Topology, Helderman Verlag."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/0165-4896(89)90042-5","article-title":"Some lifting theorems for continuous utility functions","volume":"18","author":"Herden","year":"1989","journal-title":"Math. Social Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0165-4896(90)90036-7","article-title":"On a lifting theorem of Nachbin","volume":"19","author":"Herden","year":"1990","journal-title":"Math. Social Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1155\/S0161171293000821","article-title":"Order extension of order monomorphisms on a preordered topological space","volume":"16","author":"Mehta","year":"1993","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1430","DOI":"10.1287\/moor.2020.1095","article-title":"Extension of monotonic functions and representation of preferences","volume":"46","author":"Evren","year":"2021","journal-title":"Math. Oper. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1016\/j.jmateco.2011.07.003","article-title":"On the multi-utility representation of preference relations","volume":"47","author":"Evren","year":"2011","journal-title":"J. Math. Econom."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.jmateco.2012.05.001","article-title":"Continuous multi-utility representations of preorders","volume":"48","author":"Bosi","year":"2012","journal-title":"J. Math. Econom."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s11083-006-9047-8","article-title":"On a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller","volume":"23","author":"Bosi","year":"2006","journal-title":"Order"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.mathsocsci.2015.10.006","article-title":"On continuous multi-utility representations of semi-closed and closed preorders","volume":"79","author":"Bosi","year":"2016","journal-title":"Math. Soc. Sci."},{"key":"ref_23","unstructured":"Grotemeyer, K.P. (1966). Topologie, Mathematisches Institut der Freien Universit\u00e4t."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/123\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:16:42Z","timestamp":1760120202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,27]]},"references-count":23,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020123"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020123","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,1,27]]}}}