Boolean Perspectives of Idioms and the Boyle Derivative

Abstract

We are concerned with the boolean or more generally with the complemented properties of idioms (complete upper-continuous modular lattices). Simmons (Cantor–Bendixson, socle, and atomicity. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/002-Atom.pdf, 2014) introduces a device which captures in some informal speaking how far the idiom is from being complemented, this device is the Cantor-Bendixson derivative. There exists another device that captures some boolean properties, the so-called Boyle-derivative, this derivative is an operator on the assembly (the frame of nuclei) of the idiom. The Boyle-derivative has its origins in module theory. In this investigation we produce an idiomatic analysis of the boolean properties of any idiom using the Boyle-derivative and we give conditions on a nucleus j such that [jtp] is a complete boolean algebra. We also explore some properties of nuclei j such that \(A_{j}\) is a complemented idiom.

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Acknowledgements

We would like to thank the referee for a careful and detailed reading of the manuscript and suggestions to improve it. This work was supported by the grant UNAM-DGAPA-PAPIIT IN100517.

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Correspondence to Angel Zaldívar Corichi.

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This work was supported by the Grant UNAM-DGAPA-PAPIIT IN100517.

Communicated by Jorge Picado.

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Pérez, J.C., Bárcenas, M.M., Montes, J.R. et al. Boolean Perspectives of Idioms and the Boyle Derivative. Appl Categor Struct 27, 65–84 (2019). https://doi.org/10.1007/s10485-018-9543-1

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Keywords

  • Complete boolean algebras
  • Lattices
  • Frames
  • Modules
  • Rings

Mathematics Subject Classification

  • Primary 06Cxx
  • Secondary 16S90