Equational weighted tree transformations

We consider systems of equations of weighted tree transformations with finite support over continuous and commutative semirings. We define a weighted relation to be equational, if it is a component of the least solution of such a system of equations in a pair of algebras. In particular, we focus on equational weighted tree transformations which are equational relations obtained by considering the least solutions of such systems in pairs of term algebras. We characterize equational weighted tree transformations in terms of weighted tree transformations defined by different weighted bimorphisms. To demonstrate the robustness of equational weighted tree transformations, we give an equational definition of the class of linear and nondeleting weighted top-down tree transformations and of the class of linear and nondeleting weighted extended top-down tree transformations. Finally, we prove that a weighted relation is equational if and only if it is, roughly speaking, the morphic image of a weighted equational tree transformation.