Galois Connections and Pair Algebras
Jean-Claude Derdérian · Canadian Journal of Mathematics · 1969
Unless further restricted, P, Q, and R denote arbitrary partially ordered sets whose order relations are all written “≦” . An isotone mapping ϕ: P → Q is said to be residuated if there is an isotone mapping ψ: Q → P such that (RM 1) xϕψ ≧ x for all x i n P; (RM 2) yψϕ ≦ for all y in Q. Let Q* denote the partially ordered set with order relation dual to that of Q. (A) The following conditions are equivalent: (i) ϕ: P → Q* is a Galois connection; (ii) ϕ: P → Q is a residuated mapping; (iii) Max{z ∈ P: zy ≦ y} exists for all y in Q and is equal to yψ. Since ψ is uniquely determined by ϕ, it will be denoted by ϕ+.