Reconstructing an Open Order from Its Closure, with Applications to Space-Time Physics and to Logic

  • Authors:

  • Francisco Zapata;Vladik Kreinovich

  • Affiliations:

  • Department of Computer Science, University of Texas at El Paso, El Paso, USA 79968;Department of Computer Science, University of Texas at El Paso, El Paso, USA 79968

  • Venue:
  • Studia Logica
  • Year:
  • 2012

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Abstract

In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation $${\preccurlyeq}$$ can be uniquely reconstructed if we know the "interior" $${\prec}$$ of the order relation. It is also known that in some cases, we can uniquely reconstruct $${\prec}$$ (and hence, topology) from $${\preccurlyeq}$$ . In this paper, we show that, in general, under reasonable conditions, the open order $${\prec}$$ (and hence, the corresponding topology) can be uniquely determined from its closure $${\preccurlyeq}$$ .