Let $N_1$, $N_2$ be $Q_1$, $Q_2$-subsemimodules of an $R$-semimodule $M$, with $N_1 \cap N_2 = \{0\}$. Is $N_1 + N_2$ a $Q$-subsemimodule of $M$?
I think the answer of the question is yes, but I cannot find the subset $Q$ of $M$. For more information about $Q$-subsemimodules, refer to my paper:
J. N. Chaudhari and D. R. Bonde, On Partitioning and Subtractive Subsemimodules of Semimodules over Semirings; Kyungpook Math. J. 50(2010), 329-336.