A semiring isomorphic to a cartesian product of two semirings

Question

Let $S=\lbrace 0, 1, 2, 3, 4, 5 \rbrace$, $R=\lbrace 0, 1\rbrace$ and $T=\lbrace 0, 1, 2\rbrace$, and the addition (+) and the multiplication $(\cdot)$ in $S$, $R$ and $T$ be defined by $\max$ and $\min$, respectively. Then $(S, +, \cdot)$, $(R, +, \cdot)$ and $(T, +, \cdot)$ are semirings. Also, $(R\times T, +, \cdot)$ is a semiring in which the the addition and the multiplication are defined component-wise. Note that $R\times T=\lbrace (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) \rbrace.$

How do we know that $S$ is a semiring isomorphic to $R\times T$?

Answer 1

We can't know it because it is false!

Note that $S$ satisfies the following property: for any $x,y\in S$, $x+y$ is either $x$ or $y$.

But this is not true in $R\times T$: $(1,0)+(0,2)=(1,2)$.

So they cannot be isomorphic.