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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

1 to 8 of 8

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