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Is there any smaller sort of things (without proving semiring axioms ) to ascertain that a subset $R$ of the semiring $S$ is sub-semiring ? Or, What is the necessary and sufficient condition for a subset of the semiring to be sub- semiring ?

gete
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  • For every $a$ and $b$ in the semiring, $0$, $1$, $ab$ and $a+b$ should be in the semiring. I think that it is sufficient. – ajotatxe Oct 20 '18 at 13:28
  • @ajotatxe how does distributive property inherit from the semiring? – gete Oct 20 '18 at 13:34

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