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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.

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    $\begingroup$ If $R$ is a monosemiring, then $x = x+0 = x0 = x(0+0) = (x0) + (x0) = (x+0) + (x+0) = x+x$. $\endgroup$ – Arturo Magidin Nov 5 '17 at 6:05
  • $\begingroup$ I still have a bit confusion as " Does every monosemiring posses an identity"? Because in general, some authors seem to ignore the requirement of identities in defining a semiring. here, in your answer the monosemiring R has the identity 0. would you kindly clarify it please! $\endgroup$ – gete Nov 5 '17 at 8:21
  • $\begingroup$ @gete Are you asking whether associativity and distributivity imply idempotency? In that case the answer is "no"; there is a two-element counterexample. - To make your question clear, you should define what you mean by monosemiring. (Unless you want to ask what the customary definition or notation is. But it is my impression that these algebraic structures are not very popular.) $\endgroup$ – Goldstern Nov 5 '17 at 14:54
  • $\begingroup$ @Goldstern, thank you for your concern.. Let me try to make my question more clear.. A semiring (R,+,.) is said to be an idempotent semiring if x.x = x and x+x = x∀x ∈ R and the semiring (R,+,.) is said to be monosemiring if x.y = x+y ∀x,y ∈ R. There are some authors who treat every monosemiring as a particular class of idempotent semiring. My problem was that how to prove that every monosemiring is an idempotent semiring. I agree with the answer posed by Arturo Magidin, sir. But he has used identity 0 in his proof. (continued) $\endgroup$ – gete Nov 5 '17 at 16:16
  • $\begingroup$ @Goldstern So, my next confusion is do we always assume the presence of identity in a monosemiring? because some algebraists ignores the requirement of identity elements in a semiring. Anyways, i am helped by your valuable inputs and hope for more such valuable suggestions. Thank you $\endgroup$ – gete Nov 5 '17 at 16:16

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