Let $R$ be an idempotent semiring and $ax=y$ and $by=x~\forall ~x,y\in R$, then show that $x=y$

Question

Here, a semiring $(R, +, .)$ is an idempotent in the sense that $x+x=x$ and $x.x=x~\forall~x\in R$. Let $R$ be an idempotent semiring and $ax=y$ and $by=x~\text{if }x\leq y~\text{and }y\leq x,~\text{respectively }\forall ~x,y\in R$ and some $a, b$ in $R$, then show that $x=y$. Note: Here, i am actually trying to show that the relation $\leq$ on $R$ is an anti-symmetric relation with respect to the multiplicative operation on $R$. Further i have seen that if $R$ is a multiplicatively cancellative then $x=y$ can be easily verified but i don't need the $R$ to be cancellative.