$*$-$\mu$-semirings and $*$-$\lambda$-semirings

Abstract

We introduce and study $*$-$\mu$-semirings and $*$-$\lambda$-semirings which generalize inductive $*$-semirings and weak inductive $*$-semirings, respectively. Also, we discuss the semiring of formal power series with coefficients in such a semiring and prove that the semiring of formal power series with coefficients in a weak inductive $*$-semiring [$\mu$-semiring, $\lambda$-semiring, $*$-$\lambda$-semiring] is a weak inductive $*$-semirng [$\mu$-semiring, $\lambda$-semiring, $*$-$\lambda$-semiring, respectively]. This gives a positive answer to one of Ésik and Kuich's open problems.