{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T15:50:44Z","timestamp":1765295444282,"version":"3.40.5"},"reference-count":38,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T00:00:00Z","timestamp":1703116800000},"content-version":"unspecified","delay-in-days":0,"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":[[2024,1]]},"abstract":"Abstract<\/jats:title>Monads prove to be useful mathematical tools in theoretical computer science, notably in denoting different effects of programming languages. In this paper, we investigate a type of monads which arise naturally from Keimel and Lawson\u2019s \n$\\mathbf{K}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-ification.<\/jats:p>A subcategory of \n$\\mathbf{TOP}_{\\mathbf{0}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is called of type \n$\\mathrm{K}^{*}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> if it consists of monotone convergence spaces and is of type \n$\\mathrm K$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the sense of Keimel and Lawson. Each such category induces a canonical monad \n$\\mathcal K$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on the category \n$\\mathbf{DCPO}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of dcpos and Scott-continuous maps, which is called the order-\n$\\mathbf{K}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-ification monad in this paper. First, for each category of type \n$\\mathrm{K}^{*}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we characterize the algebras of the corresponding monad \n$\\mathcal K$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> as k<\/jats:italic>-complete posets and algebraic homomorphisms as k<\/jats:italic>-continuous maps, from which we obtain that the order-\n$\\mathbf{K}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-ification monad gives the free k<\/jats:italic>-complete poset construction over the category \n$\\mathbf{POS}_{\\mathbf{d}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of posets and Scott-continuous maps. In addition, we show that all k<\/jats:italic>-complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order-K<\/jats:bold>-ification monad and conclude with the fact that each order-K<\/jats:bold>-ification monad is always commutative.<\/jats:p>","DOI":"10.1017\/s0960129523000403","type":"journal-article","created":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T06:13:33Z","timestamp":1703139213000},"page":"45-62","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["The order-K<\/b>-ification monads"],"prefix":"10.1017","volume":"34","author":[{"given":"Huijun","family":"Hou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hualin","family":"Miao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4314-7489","authenticated-orcid":false,"given":"Qingguo","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2023,12,21]]},"reference":[{"key":"S0960129523000403_ref38","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1016\/j.tcs.2010.02.020","article-title":"Dcpo-completion of posets","volume":"411","author":"Zhao","year":"2010","journal-title":"Theoretical Computer Science"},{"key":"S0960129523000403_ref27","doi-asserted-by":"crossref","first-page":"106869","DOI":"10.1016\/j.topol.2019.106869","article-title":"On well-filtered reflections of","volume":"267","author":"Shen","year":"2019","journal-title":"Topology and its Applications"},{"key":"S0960129523000403_ref21","doi-asserted-by":"crossref","unstructured":"Moggi, E. 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