2 problems

Let $S_{7}$ be the $3$-element additively idempotent semiring, and let $S_{7}^{0}$ be obtained from $S_{7}$ by adjoining a new zero element. Determine the cardinality of the interv…

Let $\mathbb{N}[x]$ be the polynomial semiring with coefficients in $\mathbb{N}$, and let $\mathbb{N}[x]=C_0\cup C_1$ be a bipartition. The polynomial semiring conjecture. There ex…