A Mezei–Wright theorem for categorical algebras

The main result of this paper is a generalization of the Mezei–Wright theorem, a result on solutions of a system of fixed point equations. In the typical setting, one solves a system of fixed point equations in an algebra equipped with a suitable partial order; there is a least element, suprema of ω-chains exist, the operations preserve the ordering and least upper bounds of ω-chains. In this setting, one solution of this kind of system is provided by least fixed points. The Mezei–Wright theorem asserts that such a solution is preserved by a continuous, order preserving algebra homomorphism.In several settings such as (countable) words or synchronization trees there is no well-defined partial order but one can naturally introduce a category by considering morphisms between the elements. The generalization of this paper consists in replacing ordered algebras by “categorical algebras”; the least element is replaced by an initial element, and suprema of ω-chains are replaced by colimits of ω-diagrams. Then the Mezei–Wright theorem for categorical algebras is that initial solutions are preserved by continuous morphisms. We establish this result for initial solutions of parametric fixed point equations.One use of the theorem is to characterize an “algebraic” element as one that can arise as a solution of some system of fixed point equations. In familiar examples, an algebraic element is one that is context-free, regular or rational. Then, if h:A→B is a continuous morphism of categorical algebras, the algebraic objects in B are those isomorphic to h-images of algebraic objects in A.