In general, the multiplicative operation in a semiring distributes over the additive operation from both the left and the right. But i found some operations which satisfy all the conditions of a semiring structure in which the multiplicative operations distributes over the additive operation and vice-versa. Infact, monosemiring is the one where both the ring opera tions distribute over each other. Can i still call the set equipped with such operations other than monosemiring a semiring?
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Such a semiring will be isomorphic to a boolean ring. It is easy to show that $a^2=a$ for all $a$ in such a semi ring. $a+0\times 0= (a+0)\times (a+0)$.
baharampuri
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your answer may be good when a semiring contains additive identity. But a semiring may not always contain additive identity. – gete May 12 '18 at 03:42
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without 0 I don't see you can say much, can you think of an example without 0 and satisfying both the distributive laws. – baharampuri May 12 '18 at 06:43
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Counter examples are available depending the operations we define and the types of the elements in the set. So far, the additive identity zero and and distributive axiom is concerned, i don't think that multiplicative absorption follows directly from distributivity in a semiring, but it does in a ring. So, distributive law holds in a semiring even without 0 element. – gete May 12 '18 at 06:56