In general, a semiring contains two distinct identities with respect to addition and multiplication defined on it. In particular, monosemiring is the one in which the neutral element is common with respect to both the operations defined on it. I found some semiring like structures other than monosemiring in which both the operations have same neutral element.

  • $\begingroup$ Part of the definition is that the additive identity is a multiplicative annihilator, so the answer to your question is no. $\endgroup$ – Hayden May 12 '18 at 4:26
  • $\begingroup$ @Hayden you have rightly pointed out but i have been wondering that what is the significance\necessity of additive identity being a multplicative annihilator. If we could show the non necessity of additive identity being a multiplicative annihilator would be an interesting result. $\endgroup$ – gete May 12 '18 at 5:20

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