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nLab > Latest Changes: rig

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  1.  
    • CommentRowNumber1.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 2nd 2022

    Added link to “multiplicatively cancellable semi-ring”.

    diff, v32, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    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.

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022
    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 3rd 2022
    • (edited Aug 3rd 2022)

    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.

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    I’ve seen four different definitions of a “semiring” out there, depending on the author:

    • A semigroup object in the category of commutative semigroups
    • A monoid object in the category of commutative semigroups
    • A semigroup object in the category of commutative monoids
    • A monoid object in the category of commutative monoids

    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

    • CommentRowNumber6.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    *define a ring to be a monoid object in the category of abelian groups

  7. adding paragraph on the relationship between rigs and semirings.

    Anonymous

    diff, v35, current

    • CommentRowNumber8.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 18th 2022

    Added a reference to Rig

    diff, v38, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2023

    added pointer to

    • William Lawvere, pp. 1 of: Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf]

    Is this (on p. 2) maybe the actual origin of the term “rig”?

    diff, v40, current

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 25th 2023

    It appears earlier than 1992, e.g., in

    • S.H. Schanuel, Negative sets have Euler characteristic and dimension, in: Proceedings of Category Theory, Como, Italy,1990, in: Lecture Notes in Mathematics, vol. 1488, Springer-Verlag, 1991, pp. 379–385.

    See here, page 379. But I wouldn’t bet this is the earliest appearance either.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2023

    Thanks, I have added that to the entry.

    diff, v41, current

1 to 11 of 11

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