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Contraexamples in difference posets and orthoalgebras

Abstract

We show that every orthoalgebra (difference orthoposet) uniquely determines a difference orthoalgebraic structure. We give examples of posets on which there exist more than one difference operation. In spite of that, every finite chain is a uniquely determined difference poset. On a difference poset there need not exist any orthoalgebraic operation, but the category of difference orthoposets is isomorphic with the category of orthoalgebras. But a difference poset which is also an orthoposet need not be a difference orthoposet. Moreover, there exist complete lattices on which there does not exist any difference operation. Finally, we show that difference operations and orthoalgebraic operations need not be extendable on a MacNeille completion of the base poset.

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Riečanová, Z., Bršel, D. Contraexamples in difference posets and orthoalgebras. Int J Theor Phys 33, 133–141 (1994). https://doi.org/10.1007/BF00671618

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Keywords

  • Field Theory
  • Elementary Particle
  • Quantum Field Theory
  • Complete Lattice
  • Difference Operation