Locally closed semirings and iteration semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. Esik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.