Syntactic Semiring and Language Equations

A classical construction assigns to any language its (ordered) syntactic monoid. Recently the author defined the so-called syntactic semiring of a language. We show here that elements of the syntactic semiring of L can be identified with transformations of a certain modification of the minimal automaton for L.The main issue here are the inequalities r(x₁, . . . , x_m) ⊆ L and equations r(x₁, . . . , x_m) = L where L is a given regular language over a finite alphabet A and r is a given regular expression over A in variables x₁, . . . , x_m. We show that the search for maximal solutions can be translated into the (finite) syntactic semiring of the language L. In such a way we are able to decide the solvability and to find all maximal solutions effectively. In fact, the last questions were already solved by Conway using his factors. The first advantage of our method is the complexity and the second one is that we calculate in a transparent algebraic structure.