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Semirings as defined on Wolfram MathWorld don’t have either an additive or multiplicative identity; they are semigroup objects in the category of commutative semigroups.
The same is true of Kazimierz Glazek’s A guide to the literature on semirings and their applications in mathematics and information sciences
Thank you, it seems very logical to me now. I will use the term rig for the structure with the two identities from now on.
I’ve seen four different definitions of a “semiring” out there, depending on the author:
The problem is already there in the definition of a ring, as some authors define a ring to be a semigroup object in the category of abelian groups, while other authors define a ring to be a monoid object in the category of commutative monoids
*define a ring to be a monoid object in the category of abelian groups
added pointer to
Is this (on p. 2) maybe the actual origin of the term “rig”?
It appears earlier than 1992, e.g., in
See here, page 379. But I wouldn’t bet this is the earliest appearance either.
Thanks to a reference provided by Rod McGuire in another thread (here):
we can settle the question of origin of the terminology ’rig’ – because Lawvere writes there, about his work with Schanuel, that:
We were amused when we finally revealed to each other that we had each independently come up with the term ’rig’.
Have added this to the entry.
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