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Seems a distributive lattice is a semiring; if a semifield is a semiring where multiplication by a non-zero is invertible, then implies assuming embeddability in a semifield. So, the answer is no.
Connes and Consani, for instance, say a semifield is a semiring (ie a rig) where the nonzero elements are all invertible.
renaming page to “additively idempotent semiring” to distinguish from the related notion of “multiplicatively idempotent semiring” for which “idempotent semiring” is also sometimes used in the literature.
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