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Boolean Perspectives of Idioms and the Boyle Derivative

  • Jaime Castro Pérez
  • Mauricio Medina Bárcenas
  • José Ríos Montes
  • Angel Zaldívar Corichi
Article

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.

Keywords

Complete boolean algebras Lattices Frames Modules Rings 

Mathematics Subject Classification

Primary 06Cxx Secondary 16S90 

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Notes

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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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Escuela de Ingeniería y CienciasInstituto Tecnolólogico y de Estudios Superiores de MonterreyMexicoMexico
  2. 2.Department of MathematicsChungnam National UniversityYuseong-gu, DaejeonRepublic of Korea
  3. 3.Instituto de MatemáticasUniversidad Nacional, Autónoma de México, Area de la Investigación CientíficaMexicoMexico
  4. 4.Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e IngenieríasUniversidad de GuadalajaraGuadalajaraMexico

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