Summary: We consider systems of equations of polynomial weighted tree transformations over the max-plus (or: arctic) semiring \(\mathbb R_{ \max } = ( \mathbb R _{ + } \cup \{ - \infty \}, \max , + , - \infty ,0)\). We apply discounting with a parameter \(0 \leq d < 1\) in order to guarantee the existence of the least solution, called least \(d\)-solution, of such systems. We compute least \(d\)-solutions under \(u\)-substitution mode, where \(u = [IO]\) or \(u = OI\). We define a weighted relation over \(\mathbb R_{\max }\) to be \(u\)-\(d\)-equational, if it is a component of the least \(u\)-\(d\)-solution of such a system of equations in a pair of algebras. We mainly focus on \(u\)-\(d\)-equational weighted tree transformations which are equational relations obtained by considering the least \(u\)-\(d\)-solutions in pairs of term algebras. We also introduce \(u\)-\(d\)-equational weighted tree languages over \(\mathbb R_{ \max }\). We characterize \(u\)-\(d\)-equational weighted tree transformations in terms of weighted tree transformations defined by weighted \(d\)-bimorphisms, which are bimorphisms from \(d\)-recognizable weighted tree languages. Finally, we prove that a weighted relation is \(u\)-\(d\)-equational if and only if it is, roughly speaking, the morphic image of a weighted \(u\)-\(d\)-equational tree transformation.
For the entire collection see [Zbl 1225.68022].