  In this paper, we study ideal- and congruence-simpleness for the Leavitt path
algebras of directed graphs with coefficients in a commutative semiring S, as
well as establish some fundamental properties of those algebras. We provide a
complete characterization of ideal-simple Leavitt path algebras with
coefficients in a semifield S that extends the well-known characterizations
when the ground semiring S is a field. Also, extending the well-known
characterizations when S is a field or commutative ring, we present a complete
characterization of congruence-simple Leavitt path algebras over row-finite
graphs with coefficients in a commutative semiring S.
