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What is the difference between a monosemiring and a semigroup?

The following definitions are for clarity of my question. A semigroup $S$ is a non empty set that satisfies closure and associativity with respect to the binary operation defined on it. A non empty set $(S, +, .)$ is said to be a semiring if both the binary operations follow closure and associativity and $(.)$ distributes over $(+)$ from both left and right. A semiring $(S, +, .)$ is said to be a monosemiring if $x.y=x+y$ for all $x,y$ in $S$. By this definition, we see that in a monosemiring there a common binary operation which is same as that of a semigroup. Moreover, it seems to be possible that every semigroup can be extended to a monosemiring by means of distributivity. What is the exact difference between monosemiring and semigroup?

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    $\begingroup$ A monosemiring require distributivity, so you must have $x+(y+z) = x(y+z) = (xy)+(xz) = x+y+x+z$. That is, for all $x,y,z$, $x+y+z = x+x+y+z$. Not every semigroup satisfies this identity, so not every semigroup can be turned into a monosemiring. $\endgroup$ – Arturo Magidin Nov 5 '17 at 6:32
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    $\begingroup$ P.S. All posts are questions; you don't need to add a "Q." at the beginning of the title. $\endgroup$ – Arturo Magidin Nov 5 '17 at 6:37
  • $\begingroup$ Thank you so much for quick response and valauble suggestions $\endgroup$ – gete Nov 5 '17 at 7:20
  • $\begingroup$ And questions should be inside the text (even if in the title). I edited accordingly $\endgroup$ – YCor Nov 5 '17 at 10:09
  • $\begingroup$ @gete: Your definition of "monosemiring" is incomplete; it's not a "non empty set" but rather a "semiring $(S,+,\cdot)$ is a monosemiring if $xy=x+y$ f\ro all $x,y\in S$". $\endgroup$ – Arturo Magidin Nov 5 '17 at 21:05

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