Invertibility of functional Galois connections
A functional Galois connection is a dual Galois connection \(B:{\mathcal F}\to {\mathcal G}\) between a sublattice \({\mathcal F}\) of \(\overline \mathbb{R}^Y\) and a sublattice \({\mathcal G}\) of \(\overline \mathbb{R}^X\), where \(X,Y\) are two sets and \(\overline\mathbb{R} =\mathbb{R}\cup \{\pm\infty\}\). The authors consider the following problem: for given \(g\in{\mathcal G}\) and \(X'\subset X\) find \(f\in {\mathcal F}\) such that \(Bf\leq g\), \(Bf(x)=g(x)\), \(\forall x\in X'\). Theorems 2 and 3 give effective conditions on \(g\) (in terms of generalized subdifferentials) for the solution \(f\) to exist and be unique, and so extend Zimmerman's covering theorem for max-plus linear equations.
scientific article; zbMATH DE number 1921081