阶 K 化单子

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-12-21 DOI:10.1017/s0960129523000403
Huijun Hou, Hualin Miao, Qingguo Li
{"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}
引用次数: 0

摘要

在理论计算机科学中,单子被证明是有用的数学工具,特别是在表示编程语言的不同效果时。在本文中,我们将研究一种由 Keimel 和 Lawson 的 $\mathbf{K}$-ification 自然产生的单子类型。如果 $\mathbf{TOP}_{\mathbf{0}}$ 的子类由单调收敛空间组成,并且在 Keimel 和 Lawson 的意义上属于 $\mathrm K$ 类型,那么这个子类就叫做 $\mathrm{K}^{*}$ 类型。每个这样的范畴都会在dcpos和斯科特连续映射的范畴$\mathbf{DCPO}$上诱导出一个典型的单体$\mathcal K$,本文称其为order-$\mathbf{K}$-ification单体。首先,对于每个$\mathrm{K}^{*}$类型的范畴,我们将相应单元$\mathcal K$的代数代数同态描述为k-完备的poset,将代数同态描述为k-连续映射,由此我们得到order-$\mathbf{K}$-ification单元给出了在poset和Scott-连续映射的范畴$\mathbf{POS}_{\mathbf{d}$上的自由k-完备poset构造。此外,我们还证明了所有 k-完备正集和斯科特连续映射构成了一个笛卡尔闭合范畴。此外,我们还考虑了阶-K-化一元体的强大性,并以每个阶-K-化一元体总是交换的这一事实得出结论。
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The order-K-ification monads

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.

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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
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.
期刊最新文献
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