{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:36:56Z","timestamp":1753889816729,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000780","name":"European Commission","doi-asserted-by":"crossref","award":["731143"],"award-info":[{"award-number":["731143"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>Given a poset $P$, the set, $\\Gamma(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\\mathbf{C}$ of $\\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $\\Gamma$-faithful if for any posets $P$ and $Q$ in $\\mathbf{C}$, $\\Gamma(P) \\cong \\Gamma(Q)$ implies $P \\cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $\\Gamma$-faithful, while $\\mathbf{Pos}_d$ is not. Ho &amp;amp; Zhao (2009) asked whether the category $\\mathbf{DCPO}$ of dcpos is $\\Gamma$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $\\Gamma$-faithful. This subcategory subsumes all currently known $\\Gamma$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.<\/jats:p><jats:p>Comment: 19 pages, 4 figures<\/jats:p>","DOI":"10.23638\/lmcs-14(1:7)2018","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:28:05Z","timestamp":1743701285000},"source":"Crossref","is-referenced-by-count":0,"title":["The Ho-Zhao Problem"],"prefix":"10.23638","volume":"Volume 14, Issue 1","author":[{"given":"Weng Kin","family":"Ho","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean","family":"Goubault-Larrecq","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Achim","family":"Jung","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyong","family":"Xi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2018,1,17]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1607.03286v3","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1607.03286v3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:28:05Z","timestamp":1743701285000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/1529"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,17]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-14(1:7)2018","relation":{"has-preprint":[{"id-type":"arxiv","id":"1607.03286v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1607.03286","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1607.03286","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2018,1,17]]},"article-number":"1529"}}