A universal mapping characterization of the completion by cuts

@article{Bishop1978AUM,
  title={A universal mapping characterization of the completion by cuts},
  author={A. A. Bishop},
  journal={algebra universalis},
  year={1978},
  volume={8},
  pages={349-353}
}
Closed maps of lattices are defined to be those satisfying the condition that the inverse image of closed ideals are closed ideas. Residuated maps are closed and closed maps are complete-join homomorphisms. The natural embeddingj of a lattice into its completion by cuts is a closed map. For every closed map ϕ from a latticeL into a complete latticeM, there exists a unique closed map φ* from the completion by cutsL intoM such that φ*j=φ. This characterizes the completion by cuts. 
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References

A Universal Mapping Property for a Lattice Completion