Is every monosemiring an idempotent semiring?
To make my question clear, let me give definitions as follows: A semiring $(R, +, .)$ is said to be monosemiring if $x.y= x+y$ for all $x, y$ in $R$. And idempotent semiring is the one in which $x.x= x=x+x$ for all $x$ in $R$. From this definition, we don't see that every monosemiring is an idempotent too. But there are some authors who define that monosemiring in an idempotent semiring satisfying the identity $x.y=x+y$ for all $x,y$ in $R$. So, from this definition, it seems like every monosemiring is an idempotent too.