Profinite completions and canonical extensions of semilattice reducts of distributive lattices
Maria João Gouveia, Hilary A. Priestley · 2013
Abstract. A bounded distributive lattice L has two unital semilattice reducts, denoted L ∧ and L∨. These ordered structures have a common canonical extension Lδ. As algebras, they also possess profinite completions, L̂, L̂ ∧ and L̂∨; the first of these is well known to coincide with Lδ. Depend-ing 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. 1.