Internal axioms for domain semirings

We investigate modal algebras that arise by adding operations of domain and antidomain to semirings and Kleene algebras. Let (S,+, ·, 0, 1) be a noncommutative and additively idempotent semiring with natural order defined as x ≤ y ⇔ x + y = x. Such semirings have many computational models, e.g., binary relations under union and relative product, languages, paths in graphs or program traces under union and complex product, actions of a (labelled transition) system under nondeterministic choice and sequential composition. In all these models, a notion of domain can be axiomatised. It charaterises, e.g., those states that a binary relation links to any other state, the starting states in a set of paths or traces, the states from which some action of a system is enabled. We provide a more abstract characterisation in the setting of semirings. Formally, a domain semiring is a semiring S extended by a domain operation d : S → S which, for all x, y ∈ S, satisfies the axioms