{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T11:53:29Z","timestamp":1785412409071,"version":"3.56.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T00:00:00Z","timestamp":1783987200000},"content-version":"unspecified","delay-in-days":194,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2026]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Whether the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0960129526100619_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper F upper S<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>F<\/mml:mi>\n                          <mml:mi>S<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$FS$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -domain of closed discs in the plane is a retract of a bifinite domain is a long-standing open problem in domain theory, which has been stated in textbooks and several other references. In this paper, we give a positive answer to this question. As preparation, we present some rules for the inductive construction of a Plotkin-poset. Then, we inductively construct a bottomed Plotkin-poset based on closed circular rings in the plane with rational radii and rational centers. Since every element in this bottomed Plotkin-poset is compact, it follows that the ideal completion of this Plotkin-poset is a bifinite domain. Furthermore, an embedding-retraction pair is established between the domain of closed discs in the plane and this bifinite domain.\n                  <\/jats:p>","DOI":"10.1017\/s0960129526100619","type":"journal-article","created":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T06:44:34Z","timestamp":1784011474000},"update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A bifinite domain based on closed circular rings"],"prefix":"10.1017","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5886-8246","authenticated-orcid":false,"given":"Min","family":"Liu","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05mxya461","id-type":"ROR","asserted-by":"publisher"}],"name":"Chang\u2019an University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,7,14]]},"reference":[{"key":"S0960129526100619_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(99)00171-6"},{"key":"S0960129526100619_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(90)90124-Z"},{"key":"S0960129526100619_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s10114-013-1240-2"},{"key":"S0960129526100619_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(83)90095-6"},{"key":"S0960129526100619_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0073967"},{"key":"S0960129526100619_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2008.06.026"},{"key":"S0960129526100619_ref14","volume-title":"Hyperspaces: Fundamentals and Recent Advances","author":"Wanes","year":"1999"},{"key":"S0960129526100619_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2019.106893"},{"key":"S0960129526100619_ref1","volume-title":"Handbook of Logic in Computer Science","author":"Abramsky","year":"1994"},{"key":"S0960129526100619_ref22","first-page":"51","article-title":"A note on the problem when \n\nupper F upper S\nFS\n$FS$\n\n\n-domains coincide with \n\nupper R upper B\nRB\n$RB$\n\n\n-domains","volume":"8","author":"Zou","year":"2018","journal-title":"Categories and General Algebraic Structures with Applications"},{"key":"S0960129526100619_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2019.07.029"},{"key":"S0960129526100619_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(00)00222-X"},{"key":"S0960129526100619_ref15","doi-asserted-by":"publisher","DOI":"10.1137\/0205035"},{"key":"S0960129526100619_ref6","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.1990.113731"},{"key":"S0960129526100619_ref16","unstructured":"Scott, D. S. (1970). Outline of a mathematical theory of computation. In: The Fourth Annual Princeton Conference on Information Sciences and Systems, Princeton, NJ, Princeton University Press, 169\u2013176."},{"key":"S0960129526100619_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511542725"},{"key":"S0960129526100619_ref13","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1951-0042109-4"},{"key":"S0960129526100619_ref4","volume-title":"Cartesian Closed Categories of Domains","author":"Jung","year":"1989"},{"key":"S0960129526100619_ref7","doi-asserted-by":"crossref","unstructured":"Jung, A. and Tix, R. (1998). The troublesome probabilistic powerdomain. In: Edalat, A. , Jung, A. , Keimel, K. and Kwiatkowska, M. (eds.) Proceedings of the Third Workshop on Computation and Approximation, vol. 13 of Electronic Notes in Theoretical Computer Science, Elsevier Science Publishers B.V., 23.","DOI":"10.1016\/S1571-0661(05)80216-6"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129526100619","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T06:44:35Z","timestamp":1784011475000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129526100619\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":19,"alternative-id":["S0960129526100619"],"URL":"https:\/\/doi.org\/10.1017\/s0960129526100619","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026]]},"assertion":[{"value":"\u00a9 The Author(s), 2026. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}],"article-number":"e22"}}