Category theoretic aspects of chain-valued frames: Part II: Applications to lattice-valued topology

  • Authors:

  • A. Pultr;S. E. Rodabaugh

  • Affiliations:

  • Department of Applied Mathematics and ITI (Institute of Theoretical Informatics), MFF Charles University, 11800 Praha 1, Czech Republic;Department of Mathematics and Statistics, Youngstown State University, Youngstown, OH 44555-3609, USA

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2008

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Abstract

This paper is Part II of a two-part series dealing with category theoretic aspects of chain-valued frames. Using the categorical properties established for L-Frm in Part I for L a complete chain, this paper constructs ''upper'' free functor L and ''lower'' free functor R. The functor L is used to create a class of non-generated L-frames, factor the LPT spectrum functor through the @S spectrum functor, resolve the relationship between L-sobriety and @i"L-sobriety, and give insight into the construction of ''universal''L-topological spaces; and the functor R is used to create a class of non-generated L-frames, factor the @w"L functor through a new spectrum functor @S^*, and give insight into the construction of ''co-universal''L-topological spaces. These facts-universal spaces are anti-stratified, co-universal spaces are stratified, L-Frm produces both kinds of spaces via L and R, L-Frm (via its dual L-Loc) is adjunctive with L-Top-comprise a coherent argument that L-Top must accommodate both kinds of spaces, resolving the philosophical debate over the place of the constant maps condition with respect to the axioms of L-topologies.