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Profinite completions and canonical extensions of semilattice reducts of distributive lattices

Gouveia, M. J.; Priestley, H. A.

Houston J. Math., 39(4) (2013), 1117-1136

A bounded distributive lattice L has two unital semilattice
reducts, denoted L? and L?. These ordered structures have a common canonical extension L?. As algebras, they also possess profinite completions, ... the first of these is well known to coincide with L . Depending on the structure of L, these three completions may coincide or may be different. Necessary and sufficient conditions are obtained for the canonical extension of L to coincide with the profinite completion of one, or of each, of its semilattice reducts. The techniques employed here draw heavily on duality theory and on results from the theory of continuous lattices.