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.
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So you define $x \le y$ if there exists $a$ such that $ax = y$? And what does it mean when you write $ax=y$ and $by=x~\forall ~x,y\in R$? – Paul Frost Nov 30 '18 at 18:19
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@Paul Frost $\leq $ is a relation defined on $R$ such that $ \forall~x, y\in R,$ $ax=y $ whever $x\leq y$ for some $a\in R$. Now, to prove that the relation $\leq$ is anti-symmetric, we have to assume that $x\leq y$ and $y\leq x$ and we need to show that $x=y$ using the relation as defined. – gete Nov 30 '18 at 18:32