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Bifinite Chu Spaces

  • Manfred Droste
  • Guo-Qiang Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4728)

Abstract

This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three base categories of Chu spaces: the generic Chu spaces (C), the extensional Chu spaces (E), and the biextensional Chu spaces (B). The main results are
  • a characterization of monics in each of the three categories;

  • existence (or the lack thereof) of colimits and a characterization of finite objects in each of the corresponding categories using monomorphisms/injections (denoted as iC, iE, and iB, respectively);

  • a formulation of bifinite Chu spaces with respect to iC;

  • the existence of universal, homogeneous Chu spaces in this category.

Unanticipated results driving this development include the fact that:
  • in C, a morphism (f,g) is monic iff f is injective and g is surjective while for E and B, (f,g) is monic iff f is injective (but g is not necessarily surjective);

  • while colimits always exist in iE, it is not the case for iC and iB;

  • not all finite Chu spaces (considered set-theoretically) are finite objects in their categories.

Chu spaces are a general framework for studying the dualities of objects and properties; points and open sets; terms and types, under rich mathematical contexts, with important connections to several sub-disciplines in computer science and mathematics. Traditionally, the study on Chu spaces had a “non-constructive” flavor. There was no framework in which to study constructions on Chu spaces with respect to their behavior in permitting a transition from finite to infinite in a continuous way and to study which constructs are continuous functors in a corresponding algebroidal category. The work presented here provides a basis for a constructive analysis of Chu spaces and opens the door to a more systematic investigation of such an analysis in a variety of settings.

References

  1. 1.
    Droste, M., Zhang, G.-Q.: Bifinite Chu spaces. In: Proc. 2nd Conference on Algebra and Coalgebra in Computer Science. LNCS, vol. 4624, pp. 179–193. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Plotkin, G.: A powerdomain construction. SIAM J. Comput. 5, 452–487 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Pratt, V.: Chu spaces. School on Category Theory and Applications, Textos Mat. SCr. B 21, 39–100 (1999)MathSciNetGoogle Scholar
  4. 4.
    Zhang, G.-Q.: Chu spaces, concept lattices, and domains. In: Proceedings of the 19th Conference on the Mathematical Foundations of Programming Semantics, Montreal, Canada, March 2003. Electronic Notes in Theoretical Computer Science, vol. 83, 17 pages (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Manfred Droste
    • 1
  • Guo-Qiang Zhang
    • 2
  1. 1.Institute of Computer Science, Leipzig University, 04158 LeipzigGermany
  2. 2.Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, Ohio 44106U.S.A.

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