All cartesian closed categories of quasicontinuous domains consist of domains

Quasicontinuity is a generalisation of Scott's notion of continuous domain, introduced in the early 80s by Gierz, Lawson and Stralka. In this paper we ask which cartesian closed full subcategories exist in qCONT, the category of all quasicontinuous domains and Scott-continuous functions. The surprising, and perhaps disappointing, answer turns out to be that all such subcategories consist entirely of continuous domains. In other words, there are no new cartesian closed full subcategories in qCONT beyond those already known to exist in CONT.To prove this, we reduce the notion of meet-continuity for dcpos to one which only involves well-ordered chains. This allows us to characterise meet-continuity by "forbidden substructures". We then show that each forbidden substructure has a non-quasicontinuous function space.

[1]  Günter Bruns.  A lemma on directed sets and chains , 1967 .

[2]  Xiaoyong Xi,et al.  The Equivalence of QRB, QFS, and Compactness for Quasicontinuous Domains , 2015, Order.

[3]  Klaus Keimel,et al.  Quasicontinuous Domains and the Smyth Powerdomain , 2013, MFPS.

[4]  Jean Goubault-Larrecq,et al.  omega-QRB-Domains and the Probabilistic Powerdomain , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[5]  A. Jung,et al.  Cartesian closed categories of domains , 1989 .

[6]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[7]  Claire Jones,et al.  Probabilistic non-determinism , 1990 .

[8]  Achim Jung,et al.  The classification of continuous domains , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[9]  G. Markowsky.  Chain-complete posets and directed sets with applications , 1976 .

[10]  Jean Goubault-Larrecq,et al.  QRB, QFS, and the Probabilistic Powerdomain , 2014, MFPS.

[11]  Péter L. Erdös,et al.  AZ-Identities and Strict 2-Part Sperner Properties of Product Posets , 2014, Order.

[12]  Proceedings of the 30th Conference on the Mathematical Foundations of Programming Semantics, MFPS 2014, Ithaca, NY, USA, June 12-15, 2014 , 2014, MFPS.

[13]  Michael B. Smyth,et al.  The Largest Cartesian Closed Category of Domains , 1983, Theor. Comput. Sci..

[14]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[15]  Hui Kou,et al.  On Meet-Continuous Dcpos , 2003 .

[16]  P. Venugopalan,et al.  Quasicontinuous posets , 1990 .

[17]  Luoshan Xu,et al.  QFS-Domains and their Lawson Compactness , 2013, Order.

[18]  N. Saheb-Djahromi,et al.  CPO'S of Measures for Nondeterminism , 1980, Theor. Comput. Sci..

[19]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.