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A motivation and generalization of scott's notion of a continuous lattice

  • George Markowsky
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 871)

Abstract

In [6], Scott introduced continuous lattices as the correct setting for an abstract theory of computation. The motivation and definition of continuous lattices was primarily in topological terms. In [7], Scott discussed continuous lattices primarily from a topological point of view. However, buried in [7] is an indication (without proof) of how to approach continuous lattices from a purely order-theoretic perspective. This order-theoretic approach seems to have escaped the notice of most computer scientists. We develop this approach in the more general setting of chain-complete posets and offer some arguments in support of the thesis that "continuous posets" (chain-complete posets with a basis) are the proper setting for an abstract theory of computation. Our definition of basis also generalizes that used by Markowsky and Rosen [5]. Finally, we discuss a number of constructions which construct posets with a basis from posets with bases.

Keywords

Local Basis Finite Subset Continuous Lattice Compact Element Infinite Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Birkoff, G. Lattice Theory, 3rd ed., Amer. Math. Soc., Providence, RI, 1967.Google Scholar
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    Crawley, P. and R. P. Dilworth, Algebraic Theory of Lattices Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.zbMATHGoogle Scholar
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    Markowsky, G., "Chain-complete Posets and Directed Sets with Applications," Algebra Universal, 6 (1976) 53–68.MathSciNetCrossRefzbMATHGoogle Scholar
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    Scott, D., "Outline of a Mathematical Theory of Computation," Proc. 4th Ann. Princeton Conf. on Inform Sci. and Systems, 1970, 169–176.Google Scholar
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    Scott, D., "Continuous Lattices," Proc. Dalhousie Conf. on Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer-Verlag, Berlin, 1972, 97–136.CrossRefGoogle Scholar
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    Vuillemin, J., Syntaxe, Semantique, et Axiomatique d'un Language de Programmation Simple, These d'etat, University of Paris, Sept. 1974.Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • George Markowsky
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown Heights

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