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The order-K-ification monads

Published online by Cambridge University Press:  21 December 2023

Huijun Hou ,
Hualin Miao  and
Qingguo Li Open the ORCID record for Qingguo Li [Opens in a new window]
Huijun Hou
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan, China
Hualin Miao
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan, China
Qingguo Li*
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan, China
*
Corresponding author: Qingguo Li; Email: liqingguoli@aliyun.com
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Abstract

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.

Keywords

dcposorder-K-ification monadEilenberg-Moore algebrask-complete posets
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 34 , Issue 1 , January 2024 , pp. 45 - 62
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

*This work is supported by the National Natural Science Foundation of China (No.12231007).

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