Semiring $(S, \oplus, \otimes, \bar{0}, \bar{1})$ is an algebraic structure in which $(S, \oplus, \otimes)$ is a monoid and $(S, \otimes, \bar{1})$ is a semigroup and multiplication $\otimes$ distributes over addition $\oplus$. In general, it is noted that $\bar{0}\neq \bar{1}$ (except in case of monosemiring). I have a following counter example of a non- monosemiring in which $\bar{0}= \bar{1}$.
Let $S$ be the set of all simple undirected graphs. Then the structure $(S, \cup, \nabla, (\emptyset, \emptyset))$ is a semiring in which the empty graph $(\emptyset, \emptyset)$ is a neutral element with respect to both addition $\cup$ (graph union) and multiplication $\nabla$ (graph join). That is, in this case additive and multiplicative identity of the semiring coincides. This semiring is very similar to max- plus semiring except that $\bar{0}=\bar{1}$ here.
I look forward for any suggestions/agreement or disagreement to the above statement.