$phi$-primary subsemimodules

Let $R$ be a commutative semiring with identity and $M$ be a unitary $R$-semimodule. ##Let $phi:mathcal{S}(M)rightarrow mathcal{S}(M)cup{emptyset}$ be a function, where $mathcal{S}(M)$ is the set of all subsemimodules of $M$. ##A proper subsemimodule $N$ of $M$ is called $phi$-primary subsemimodule, if whenever $rin R$ and $xin M$ with $rxin N-phi(N)$, implies that $rin sqrt{(N:_R M)}$ or $xin N$. So if we take $phi(N)=emptyset$ (resp., $phi(N)={0}$), a $phi$-primary subsemimodule is primary (resp., weakly primary). In this paper, we study the concept of $phi$-primary subsemimodule which is a generalization of $phi$-prime subsemimodule in a commutative semiring