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Lattice-Valued Frames, Functor Categories, And Classes Of Sober Spaces

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Topological and Algebraic Structures in Fuzzy Sets

Part of the book series: Trends in Logic ((TREN,volume 20))

Abstract

This chapter introduces lattice-valued frames or L- frames, related to traditional frames analogously to how L-topological spaces relate to traditional spaces, via level sets and level mappings viewed as systems of frame morphisms (Proposition 3.3.2). En route, the well-known S2 and LS2 functors [25, 10, 11, 12, 16, 17, 18, 29, 33, 43, 44, 48, 54, 55, 56, 57, 58, 61] relating traditional spaces and L-spaces, respectively, to their associated (semi)locales of open and L-open sets are analogized and modified. This study both gives new descriptions of classes of sober spaces extant in the literature and creates a new class of sober spaces, justifying examples for which are given in Chapter 17 [42] of this Volume.

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Authors
  1. A. Pultr
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  2. S. E. Rodabaugh
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Youngstown State University, 44555-3609, Youngstown, Ohio, USA

    Stephen Ernest Rodabaugh

  2. Institut für Algebra, Stochastik und wissensbasierte mathematische Systeme, Johannes Kepler Universität, 4040, Linz, Austria

    Erich Peter Klement

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Pultr, A., Rodabaugh, S.E. (2003). Lattice-Valued Frames, Functor Categories, And Classes Of Sober Spaces. In: Rodabaugh, S.E., Klement, E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets. Trends in Logic, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0231-7_7

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  • DOI: https://doi.org/10.1007/978-94-017-0231-7_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6378-6

  • Online ISBN: 978-94-017-0231-7

  • eBook Packages: Springer Book Archive

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