{"title":"阶 K 化单子","authors":"Huijun Hou, Hualin Miao, Qingguo Li","doi":"10.1017/s0960129523000403","DOIUrl":null,"url":null,"abstract":"<p>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’s <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf{K}$</span></span></img></span></span>-ification.</p><p>A subcategory of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf{TOP}_{\\mathbf{0}}$</span></span></img></span></span> is called of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathrm{K}^{*}$</span></span></img></span></span> if it consists of monotone convergence spaces and is of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathrm K$</span></span></img></span></span> in the sense of Keimel and Lawson. Each such category induces a canonical monad <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathcal K$</span></span></img></span></span> on the category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf{DCPO}$</span></span></img></span></span> of dcpos and Scott-continuous maps, which is called the order-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf{K}$</span></span></img></span></span>-ification monad in this paper. First, for each category of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathrm{K}^{*}$</span></span></img></span></span>, we characterize the algebras of the corresponding monad <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathcal K$</span></span></img></span></span> as <span>k</span>-complete posets and algebraic homomorphisms as <span>k</span>-continuous maps, from which we obtain that the order-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf{K}$</span></span></img></span></span>-ification monad gives the free <span>k</span>-complete poset construction over the category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline11.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathbf{POS}_{\\mathbf{d}}$</span></span></span></span> of posets and Scott-continuous maps. In addition, we show that all <span>k</span>-complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order-<span>K</span>-ification monad and conclude with the fact that each order-<span>K</span>-ification monad is always commutative.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"238 1 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The order-K-ification monads\",\"authors\":\"Huijun Hou, Hualin Miao, Qingguo Li\",\"doi\":\"10.1017/s0960129523000403\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>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’s <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{K}$</span></span></img></span></span>-ification.</p><p>A subcategory of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{TOP}_{\\\\mathbf{0}}$</span></span></img></span></span> is called of type <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm{K}^{*}$</span></span></img></span></span> if it consists of monotone convergence spaces and is of type <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm K$</span></span></img></span></span> in the sense of Keimel and Lawson. Each such category induces a canonical monad <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathcal K$</span></span></img></span></span> on the category <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{DCPO}$</span></span></img></span></span> of dcpos and Scott-continuous maps, which is called the order-<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline7.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{K}$</span></span></img></span></span>-ification monad in this paper. First, for each category of type <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm{K}^{*}$</span></span></img></span></span>, we characterize the algebras of the corresponding monad <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathcal K$</span></span></img></span></span> as <span>k</span>-complete posets and algebraic homomorphisms as <span>k</span>-continuous maps, from which we obtain that the order-<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline10.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{K}$</span></span></img></span></span>-ification monad gives the free <span>k</span>-complete poset construction over the category <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline11.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{POS}_{\\\\mathbf{d}}$</span></span></span></span> of posets and Scott-continuous maps. In addition, we show that all <span>k</span>-complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order-<span>K</span>-ification monad and conclude with the fact that each order-<span>K</span>-ification monad is always commutative.</p>\",\"PeriodicalId\":49855,\"journal\":{\"name\":\"Mathematical Structures in Computer Science\",\"volume\":\"238 1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Structures in Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1017/s0960129523000403\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/s0960129523000403","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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’s $\mathbf{K}$-ification.
A subcategory of $\mathbf{TOP}_{\mathbf{0}}$ is called of type $\mathrm{K}^{*}$ if it consists of monotone convergence spaces and is of type $\mathrm K$ in the sense of Keimel and Lawson. Each such category induces a canonical monad $\mathcal K$ on the category $\mathbf{DCPO}$ of dcpos and Scott-continuous maps, which is called the order-$\mathbf{K}$-ification monad in this paper. First, for each category of type $\mathrm{K}^{*}$, we characterize the algebras of the corresponding monad $\mathcal K$ as k-complete posets and algebraic homomorphisms as k-continuous maps, from which we obtain that the order-$\mathbf{K}$-ification monad gives the free k-complete poset construction over the category $\mathbf{POS}_{\mathbf{d}}$ of posets and Scott-continuous maps. In addition, we show that all k-complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order-K-ification monad and conclude with the fact that each order-K-ification monad is always commutative.
期刊介绍:
Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.