On ∗–λ-semirings

A ∗–λ-semiring is an ordered semiring S equipped with a star operation such that for any a, b ∈ S, a∗b is the least fixed point of the linear mapping x ↦ ax + b over S. The notion of ∗–λ-semirings is a generalization of several important concepts such as continuous semirings, (weak) inductive ∗-semirings and the Kleene algebras of Conway and Kozen. We investigate several basic properties of ∗–λ-semirings and obtain results on ∗–λ-semirings in relation to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on inductive ∗-semirings.