Volume 2 - Proceedings of ISDT 9

Proceedings of the 9th International Symposium of Domain Theory and its Applications (ISDT 2022) held on a virtual platform from 4th to 6th July 2022 at the Nanyang Technological University, National Institute of Education, Singapore. DOI: [10.46298/entics.proceedings.isdt9](https://doi.org/10.46298/entics.proceedings.isdt9)

1. The $d^{*}$-space

Xiangping Chu ; Qingguo Li.
In this paper, we introduce the concept of $d^{\ast}$-spaces. We find that strong $d$-spaces are $d^{\ast}$-spaces, but the converse does not hold. We give a characterization for a topological space to be a $d^{\ast}$-space. We prove that the retract of a $d^{\ast}$-space is a $d^{\ast}$-space. We obtain the result that for any $T_{0}$ space $X$ and $Y$, if the function space $TOP(X,Y)$ endowed with the Isbell topology is a $d^{\ast}$-space, then $Y$ is a $d^{\ast}$-space. We also show that for any $T_{0}$ space $X$, if the Smyth power space $Q_{v}(X)$ is a $d^{\ast}$-space, then $X$ is a $d^{\ast}$-space. Meanwhile, we give a counterexample to illustrate that conversely, for a $d^{\ast}$-space $X$, the Smyth power space $Q_{v}(X)$ may not be a $d^{\ast}$-space.

2. Continuous R-valuations

Jean Goubault-Larrecq ; Xiaodong Jia.
We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Théron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $\tau$-smooth measure $\mu$ (defined on the Borel $\sigma$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $\mu$, when the support of $\mu$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit […]

3. Quasiexact posets and the moderate meet-continuity

Zhaorong He ; Zhongqiang Yang ; Dongsheng Zhao.
The study of weak domains and quasicontinuous domains leads to the consideration of two types generalizations of domains. In the current paper, we define the weak way-below relation between two nonempty subsets of a poset and quasiexact posets. We prove some connections among quasiexact posets, quasicontinuous domains and weak domains. Furthermore, we introduce the weak way-below finitely determined topology and study its links to Scott topology and the weak way-below topology first considered by Mushburn. It is also proved that a dcpo is a domain if it is quasiexact and moderately meet continuous with the weak way-below relation weakly increasing.

4. Weakly meet $s_{Z}$-continuity and $\delta_{Z}$-continuity

Huijun Hou ; Qingguo Li.
Based on the concept of weakly meet $s_{Z}$-continuouity put forward by Xu and Luo in \cite{qzm}, we further prove that if the subset system $Z$ satisfies certain conditions, a poset is $s_{Z}$-continuous if and only if it is weakly meet $s_{Z}$-continuous and $s_{Z}$-quasicontinuous, which improves a related result given by Ruan and Xu in \cite{sz}. Meanwhile, we provide a characterization for the poset to be weakly meet $s_{Z}$-continuous, that is, a poset with a lower hereditary $Z$-Scott topology is weakly meet $s_{Z}$-continuous if and only if it is locally weakly meet $s_{Z}$-continuous. In addition, we introduce a monad on the new category $\mathbf{POSET_{\delta}}$ and characterize its $Eilenberg$-$Moore$ algebras concretely.

5. On $k$-ranks of topological spaces

Mengjie Jin ; Qingguo Li.
In this paper, the concepts of $K$-subset systems and $k$-well-filtered spaces are introduced, which provide another uniform approach to $d$-spaces, $s$-well-filtered spaces (i.e., $\mathcal{U}_{S}$-admissibility) and well-filtered spaces. We prove that the $k$-well-filtered reflection of any $T_{0}$ space exists. Meanwhile, we propose the definition of $k$-rank, which is an ordinal that measures how many steps from a $T_{0}$ space to a $k$-well-filtered space. Moreover, we derive that for any ordinal $\alpha$, there exists a $T_{0}$ space whose $k$-rank equals to $\alpha$. One immediate corollary is that for any ordinal $\alpha$, there exists a $T_{0}$ space whose $d$-rank (respectively, $wf$-rank) equals to $\alpha$.

6. Quantaloidal Completions of Order-enriched Categories and Their Applications

Min Liu ; Yulin Li.
By introducing the concept of quantaloidal completions for an order-enriched category, relationships between the category of quantaloids and the category of order-enriched categories are studied. It is proved that quantaloidal completions for an order-enriched category can be fully characterized as compatible quotients of the power-set completion. As applications, we show that a special type of injective hull of an order-enriched category is the MacNeille completion; the free quantaloid over an order-enriched category is the Down-set completion.

7. A note on the category of c-spaces

Z. Lyu ; X. Xie ; H. Kou.
We prove that the category of c-spaces with continuous maps is not cartesian closed. As a corollary the category of locally finitary compact spaces with continuous maps is also not cartesian closed.

8. One-step closure, weak one-step closure and meet continuity

Hualin Miao ; Qingguo Li ; Dongsheng Zhao.
This paper studies the weak one-step closure and one-step closure properties concerning the structure of Scott closures. We deduce that every quasicontinuous domain has weak one-step closure and show that a quasicontinuous poset need not have weak one-step closure. We also constructed a non-continuous poset with one-step closure, which gives a negative answer to an open problem posed by Zou et al.. Finally, we investigate the relationship between weak one-step closure property and one-step closure property and prove that a poset has one-step closure if and only if it is meet continuous and has weak one-step closure.

9. $SI_2$-quasicontinuous spaces

Xiaojun Ruan ; Xiaoquan Xu.
In this paper, as a common generalization of $SI_{2}$-continuous spaces and $s_{2}$-quasicontinuous posets, we introduce the concepts of $SI_{2}$-quasicontinuous spaces and $\mathcal{GD}$-convergence of nets for arbitrary topological spaces by the cuts. Some characterizations of $SI_{2}$-quasicontinuity of spaces are given. The main results are: (1) a space is $SI_{2}$-quasicontinuous if and only if its weakly irreducible topology is hypercontinuous under inclusion order; (2) A $T_{0}$ space $X$ is $SI_{2}$-quasicontinuous if and only if the $\mathcal{GD}$-convergence in $X$ is topological.

10. Partial frames, their free frames and their congruence frames

Anneliese Schauerte ; John Frith.
The context of this work is that of partial frames; these are meet-semilattices where not all subsets need have joins. A selection function, S, specifies, for all meet-semilattices, certain subsets under consideration, which we call the ``designated'' ones; an S-frame then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, sigma-frames, kappa-frames and frames. We consider right and left adjoints of S-frame maps, as a prelude to the introduction of closed and open maps. Then we look at what might be an appropriate notion of Booleanness for partial frames. The obvious candidate is the condition that every element be complemented; this concept is indeed of interest, but we pose three further conditions which, in the frame setting, are all equivalent to it. However, in the context of partial frames, the four conditions are distinct. In investigating these, we make essential use of the free frame over a partial frame and the congruence frame of a partial frame. We compare congruences of a partial frame, technically called S-congruences, with the frame congruences of its free frame. We provide a natural transformation for the situation and also consider right adjoints of the frame maps in question. We characterize the case where the two congruence frames […]

11. Further studies on open well-filtered spaces

Chong Shen ; Xiaoyong Xi ; Dongsheng Zhao.
The open well-filtered spaces were introduced by Shen, Xi, Xu and Zhao to answer the problem whether every core-compact well-filtered space is sober. In the current paper we explore further properties of open well-filtered spaces. One of the main results is that if a space is open well-filtered, then so is its upper space (the set of all nonempty saturated compact subsets equipped with the upper Vietoris topology). Some other properties on open well-filtered spaces are also studied.

12. Strongly Continuous Domains

Yinglong Song ; Jinbo Yang.
Strong Scott topology introduced by X. Xu and D. Zhao is a kind of new topology which is finer than upper topology and coarser than Scott topology. Inspired by the topological characterizations of continuous domains and hypercontinuous domains, we introduce the concept of strongly continuous domains and investigate some properties of strongly continuous domains. In particular, we give the definition of strong way-below relation and obtain a characterization of strongly continuous domains via the strong way-below relation. We prove that the strong way-below relation on a strongly continuous domain satisfies the interpolation property, and clarify the relationship between strongly continuous domains and continuous domains, and the relationship between strongly continuous domains and hypercontinuous domains. We discuss the properties of strong Scott topology and strong Lawson topology, which is the common refinement of the strong Scott topology and the lower topology, on a strongly continuous domain.

13. Representations of Domains via CF-approximation Spaces

Guojun Wu ; Luoshan Xu.
Representations of domains mean in a general way representing a domain as a suitable family endowed with set-inclusion order of some mathematical structures. In this paper, representations of domains via CF-approximation spaces are considered. Concepts of CF-approximation spaces and CF-closed sets are introduced. It is proved that the family of CF-closed sets in a CF-approximation space endowed with set-inclusion order is a continuous domain and that every continuous domain is isomorphic to the family of CF-closed sets of some CF-approximation space endowed with set-inclusion order. The concept of CF-approximable relations is introduced using a categorical approach, which later facilitates the proof that the category of CF-approximation spaces and CF-approximable relations is equivalent to that of continuous domains and Scott continuous maps.

14. A Hofmann-Mislove theorem for $c$-well-filtered spaces

Liping Zhang ; Xiangnan Zhou ; Qingguo Li.
The Hofmann-Mislove theorem states that in a sober space, the nonempty Scott open filters of its open set lattice correspond bijectively to its compacts saturated sets. In this paper, the concept of $c$-well-filtered spaces is introduced. We show that a retract of a $c$-well-filtered space is $c$-well-filtered and a locally Lindelöf and $c$-well-filtered $P$-space is countably sober. In particular, we obtain a Hofmann-Mislove theorem for $c$-well-filtered spaces.

15. The sheaf representation of residuated lattices

Huarong Zhang ; Dongsheng Zhao.
The residuated lattices form one of the most important algebras of fuzzy logics and have been heavily studied by people from various different points of view. Sheaf presentations provide a topological approach to many algebraic structures. In this paper, we study the topological properties of prime spectrum of residuated lattices, and then construct a sheaf space to obtain a sheaf representation for each residuated lattice.