Locally Closed Semirings

Abstract. We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + ak =1 + a + ⋯ + ak + 1. In any locally closed semiring we may define a star operation a ↦ a*, where a* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring.