{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T18:48:06Z","timestamp":1767898086184,"version":"3.49.0"},"reference-count":17,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No.12231007"],"award-info":[{"award-number":["No.12231007"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No.12571495"],"award-info":[{"award-number":["No.12571495"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,1,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Let $\\textbf{Lat}_{\\lor }$ denote the category of lattices with morphisms preserving supremum of non-empty finite subsets. We prove the following:<\/jats:p>\n                  <jats:p>(1) There exists a distributive law of the ordered-ideal monad over the ordered-filter monad on $\\textbf{Lat}_{\\lor }$.<\/jats:p>\n                  <jats:p>(2) The category ${\\textbf{Cnt}}^{\\mathbb{co}}$ of cocontinuous lattices with morphisms preserving arbitrary non-empty sups and filtered infs is strictly monadic over $\\textbf{Lat}_{\\lor }$.<\/jats:p>\n                  <jats:p>(3) The category ${\\textbf{MCnt}}^{\\mathbb{co}}$ of meet-continuous and cocontinuous lattices with morphisms preserving arbitrary non-empty sups and filtered infs is isomorphic to a full subcategory of ${\\textbf{Lat}_{\\lor }}^{\\mathbb{FilId}}$, consisting of $\\mathbb{FilId}$-algebras and $\\mathbb{FilId}$-homomorphisms on ${\\textbf{Lat}_{\\lor }}$.<\/jats:p>\n                  <jats:p>For bounded-complete posets, analogous Eilenberg\u2013Moore algebras provide a categorical representation. Finally, using the concept of bounded down-sets, we derive an algebraic characterization of ${{\\textbf{Cnt}}}^{\\mathrm{co}}$.<\/jats:p>","DOI":"10.1093\/logcom\/exaf074","type":"journal-article","created":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T12:50:09Z","timestamp":1763470209000},"source":"Crossref","is-referenced-by-count":0,"title":["Algebraic representations of some kinds of cocontinuous lattices"],"prefix":"10.1093","volume":"36","author":[{"given":"Chengkai","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics, Hunan University , Changsha 410082,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingguo","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hunan University , Changsha 410082,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiangnan","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hunan 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