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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Countably QC-approximating posets.

Xuxin Mao1, Luoshan Xu2

  • 1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China.

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Summary

This study introduces countably QC-approximating posets, a generalization of countably C-approximating posets. These properties are used to characterize generalized completely distributive lattices and posets, advancing lattice theory.

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Area of Science:

  • Order Theory
  • Lattice Theory
  • Set Theory

Background:

  • Introduces countably QC-approximating posets as a generalization of countably C-approximating posets.
  • Builds upon existing concepts in order theory and lattice theory to explore new poset properties.

Purpose of the Study:

  • To introduce and define the concept of countably QC-approximating posets.
  • To characterize generalized completely distributive lattices and generalized countably approximating posets using this new property.

Main Methods:

  • Introduced the concept of countably QC-approximating posets.
  • Utilized the countably QC-approximating property to establish characterizations.
  • Investigated properties of generalized completely distributive lattices and generalized countably approximating posets.

Main Results:

  • A complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating.
  • A poset L with countably directed joins is generalized countably approximating if and only if the lattice of all σ-Scott-closed subsets of L is weakly generalized countably approximating.

Conclusions:

  • The study provides key characterizations for generalized completely distributive lattices and generalized countably approximating posets.
  • The introduced concept of countably QC-approximating posets offers a valuable tool for further research in order theory and lattice theory.