What is the difference between a monosemiring and a semigroup?

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?

• 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. – Arturo Magidin Nov 5 '17 at 6:32
• P.S. All posts are questions; you don't need to add a "Q." at the beginning of the title. – Arturo Magidin Nov 5 '17 at 6:37
• Thank you so much for quick response and valauble suggestions – gete Nov 5 '17 at 7:20
• And questions should be inside the text (even if in the title). I edited accordingly – YCor Nov 5 '17 at 10:09
• @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 $x,y\in S$". – Arturo Magidin Nov 5 '17 at 21:05