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Quasicontinuous Domains and the Smyth Powerdomain

Reinhold Heckmann, Klaus Keimel
2013 Electronical Notes in Theoretical Computer Science  
It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets.  ...  In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain.  ...  In this paper we intend to show that quasicontinuous domains should be tied up with the Smyth or upper powerdomain [15, 16] rather than the lower powerdomain.  ... 
doi:10.1016/j.entcs.2013.09.015 fatcat:i22t6pzo6zd4rcozu2z7fpawoe

QC-continuity of posets and the Hoare powerdomain of QFS-domains

Qingyu He, Luoshan Xu, Lingyun Yang
2016 Topology and its Applications  
(resp., quasialgebraic) iff the Hoare powerdomain H(L) is quasicontinuous (resp., quasialgebraic); (3) the Hoare powerdomain H(L) of a QFS-domain L is still a QFS-domain.  ...  With this new concept, the following results are obtained: (1) a complete lattice is generalized completely distributive (GCD) iff it is quasicontinuous and QC-continuous; (2) a dcpo L is quasicontinuous  ...  Acknowledgement Thanks to an anonymous editor and an anonymous referee. Their valuable comments broaden the view and improve the presentation of this paper.  ... 
doi:10.1016/j.topol.2015.10.007 fatcat:qx3qiabderhf5myzh6myzhbldm

Observationally-induced algebras in Domain Theory

Thomas Streicher, Ingo Battenfeld, Klaus Keimel, Martín Escardó
2014 Logical Methods in Computer Science  
We analyze this "deviation" and show how to reobtain the original Plotkin powerdomain by imposing further conditions previously considered by R.~Heckmann and J.~Goubault-Larrecq.  ...  ambient category is given by classical domain theory.  ...  Lawson and Xi Xiaoyong, the generalizations of FS-domains and retracts of bifinite domains to QFS-and QRB-domains agree with the compact coherent quasicontinuous dcpos, and our previous proposition applies  ... 
doi:10.2168/lmcs-10(3:18)2014 fatcat:cgpwkniipzdr7k6f7aazqrcgz4

omega-QRB-Domains and the Probabilistic Powerdomain

Jean Goubault-Larrecq
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
Is there any cartesian-closed category of continuous domains that would be closed under Jones and Plotkin's probabilistic powerdomain construction?  ...  We show that they form a category QRB with pleasing properties: QRB is closed under the probabilistic powerdomain functor, has all finite products, all bilimits, and is stable under retracts, and even  ...  Let ( ) be the Smyth powerdomain of , i.e., the poset of all non-empty compact saturated subsets of , ordered by ⊇. ( ) is a continuous dcpo whenever is locally compact and sober, with least upper bounds  ... 
doi:10.1109/lics.2010.50 dblp:conf/lics/Goubault-Larrecq10 fatcat:xplyvmen5jhrbie7gkm2kjyr4m

Coincidence of the upper Vietoris topology and the Scott topology [article]

Xiaoquan Xu, Zhongqiang Yang
2020 arXiv   pre-print
For the Alexandroff double circle Y, which is Hausdorff and first-countable, we show that its Smyth power space P_S(Y) is not first-countable.  ...  X and the continuity of (X).  ...  Keimel, Quasicontinuous domains and the Smyth powerdomain, Electronic Notes in Theor. Comp. Sci. 298 (2013) 215-232. [10] K. Hofmann and J.  ... 
arXiv:2003.06542v1 fatcat:srtuxm3corastdwijrxniezahe

On Scott power spaces [article]

Xiaoquan Xu, Xinpeng Wen, Xiaoyong Xi
2022 arXiv   pre-print
For a T_0 space X, let 𝖪(X) be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order.  ...  Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces and Scott power spaces are also investigated.  ...  The three main such powerdomains are the Smyth powerdomain for demonic non-determinism, the Hoare powerdomain for angelic non-determinism, and the Plotkin powerdomain for erratic non-determinism.  ... 
arXiv:2207.08720v1 fatcat:g7yeb5qn6fbclj6ug3j5zyhrhe

An upper power domain construction in terms of strongly compact sets [chapter]

Reinhold Heckmann
1992 Lecture Notes in Computer Science  
B Y , f A] is the image of A and f ?1 B] the inverse image of B. Domains A poset (partially ordered set) is a set P together with a re exive, antisymmetric, and transitive relation' '.  ...  Its power domains contain less elements than the classical ones in terms of compact sets, but still admit all necessary operations, i.e. they contain less junk.  ...  Acknowledgements I am most grateful to Fritz M uller, Helmut Seidl, Andreas Hense, and Christian Fecht for many fruitful discussions and careful draft reading.  ... 
doi:10.1007/3-540-55511-0_14 fatcat:hrygn4roh5d45glr4v23gzd4qe

Two topologies on the lattice of Scott closed subsets [article]

Yu Chen, Hui Kou, Zhenchao Lyu
2021 arXiv   pre-print
In this paper, we discuss the lower Vietoris topology and the Scott topology on Γ(P) and give some sufficient conditions to make the two topologies equal.  ...  For a poset P, let σ(P) and Γ(P) respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set Σ P=(P,σ(P)).  ...  Acknowledgement We thank the anonymous referee for improving the presentation of the paper. References  ... 
arXiv:2103.15139v1 fatcat:gyuudym57bdtrhpo4235cvw5ky

Book report

2004 Computers and Mathematics with Applications  
IV-8 powerdomains. The Hoare powerdomaln. The Smyth powerdomain. The Plotkin powerdomain. Exercises. New notes. IV-9 The Extended Probabilistic powerdomaln. Exercises. New notes. V.  ...  III-3 Quasicontinuity and Liminf Convergence. Exercises. Old notes. New notes. III-4 Bases and Weights. Exercises. Old notes. New Notes. IIL5 Compact Domains. Exercises. New notes. IV.  ... 
doi:10.1016/s0898-1221(04)84016-1 fatcat:szz7sksvhnentlcyspkplxuaya

Domain Theory and its Applications

Stephen Brookes, Manfred Droste, Michael Mislove
unpublished
Smyth (Imperial), was used to devise a uniform approach to the theory of continuous domains in analogy to the traditional approach to algebraic domains.  ...  More importantly, the Dagstuhl meeting was used to assess the state of domain theory, and important insights were gained through the seminar into the status and future possibilities for domain theory and  ...  Smyth (Imperial ), was used to devise a uniform approach to the theory of continuous domains in analogy to the traditional approach to algebraic domains. • Domain theory and topology: Topological aspects  ... 
fatcat:uq7x263dg5akzk2syfbubbdghq

Dagstuhl Seminar

Ralpf Kopperman, Michael Smyth, Dieter Spreen, Julian Webster
unpublished
During the seminar, several participants presented their current research, and ongoing work and open problems were discussed.  ...  Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper.  ...  the entire category of stably compact spaces and continuous maps, whereas the Smyth powerdomain is only free in the more limited sense.  ... 
fatcat:lwjzqrsfdvb3phqpdw7cvxcqoy

06341 Abstracts Collection – Computational Structures for Modelling Space, Time and Causality

Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, Dieter Spreen
2007
During the seminar, several participants presented their current research, and ongoing work and open problems were discussed.  ...  From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 "Computational Structures for Modelling Space, Time and Causality" was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl  ...  An important example of such spaces are domains in the sense of Scott. As has indeed been shown by M. B. Smyth, ω-continuous domains are partial metrizable.  ... 
doi:10.4230/dagsemproc.06341.1 fatcat:p6yl72ueoncztj3umdwbhaskey