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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

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023

    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.

    diff, v42, current

  13. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v47, current

    • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeNov 10th 2024

    For some reason this passage wasn’t showing in the definition of rig:

    and also the absorption/annihilation laws, which are the nullary version of the distributive laws:

    0⋅x=0=x⋅0

    I’m trying to fix that.

    diff, v48, current

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeNov 10th 2024

    The problem was apparently a comma directly after a double dollar sign! Removing this made the following text appear.

    diff, v48, current

  16. Adding references

    • Manfred Droste, Werner Kuich, Semirings and Formal Power Series (2009). In: Manfred Droste, Werner Kuich, Heiko Vogler (editors), Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. pp. 3–28.(doi:10.1007/978-3-642-01492-5_1)

    • Zoltán Ésik, Iteration semirings (2008). In: Masami Ito, Masafumi Toyama (editors), Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings. Lecture Notes in Computer Science. Vol. 5257. Berlin: Springer-Verlag. pp. 1–20. (doi:10.1007/978-3-540-85780-8_1)

    Anonymouse

    diff, v49, current

  17. Adding reference

    Anonymouse

    diff, v49, current

1 to 17 of 17

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