The structure of pseudocomplemented distributive lattices. Chajda I, Eigenthaler G, Länger H (2012) Congruence Classes in Universal Algebra.An extension of relative pseudocomplementation to non-distributive lattices. From a ≤ e ≤ f we have a ≤ f which implies Θ ≤ ′ Θ = Θ by (i), proving transitivity of ≤ ′. Then, by (ii) there exists some e ∈ Θ with a ≤ e and because of Θ = Θ ≤ ′ Θ some f ∈ Θ with e ≤ f. Finally, let c ∈ P and assume Θ ≤ ′ Θ and Θ ≤ ′ Θ. Therefore Θ = Θ = Θ which proves antisymmetry of ≤ ′. Since a ≤ c ≤ d, a, d ∈ Θ and ( Θ, ≤ ′ ) is convex we conclude c ∈ Θ. Because of Θ = Θ ≤ ′ Θ there exists some d ∈ Θ with c ≤ d. Then, by (ii), there exists some c ∈ Θ with a ≤ c. If, conversely, there exists some c ∈ Θ with a ≤ c, then according to (i) we have Θ ≤ ′ Θ = Θ.
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