Let $R$ be a multiplicatively idempotent semiring with additive identity, and a partial order relation $\leq$ is defined on $R$. Then, for all $x$ in $R$, does the identity $x+2x=2x$ implies $x\leq 2x$?.
I think the last line will hold if second last line holds. Please confirm.
Counterexample. Let $R = (\{0, 1, 2\}, \oplus, \otimes)$ be the semiring defined by $x \oplus y = \min \{x + y, 2\}$ and $x \otimes y = \min \{xy, 2\}$. Then $R$ is multiplicatively idempotent and the equality is a partial order relation on $R$. However $1 \oplus (1 \oplus 1) = (1 \oplus 1)$ but $1 \not= 1 \oplus 1$.