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

    Definition of a multiplicatively cancellable semi-ring.

    v1, current

  1. semirings don’t necessarily have an additive identity element 0 so the objects referred in this article are rigs rather than semirings.

    Also, the current definition is only valid in classical mathematics, in constructive mathematics more care needs to be taken with the definitions.

    Anonymous

    diff, v2, current

  2. rename somehow didn’t go through the first time

    Anonymous

    diff, v2, current

  3. rename didn’t go through again for some reason

    Anonymous

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 3rd 2022
    • (edited Aug 3rd 2022)

    “semirings don’t necessarily have an additive identity element 0 so the objects referred in this article are rigs rather than semirings” : Do you have references for this?

    Semirings are defined with an identity element 0 in “Semirings and their Applications”, Golan, 1999.

    I always use semirings, semi-rings and rigs as synonymous. I don’t know of such a distinction. Semirings and rigs are the same thing on Wikipedia and the nlab.

    I just admit that I’ve written “semi-rings” instead of “semirings” and that this is the second which is usually used.

    Please don’t use this convention without more explanations. We must keep a coherence between the different pages of the nlab.

    And if you want to defend your point, please do it on the discussion of the page “rig”, it would be more appropriate.

    • CommentRowNumber6.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    The established terminology used in the nLab is to refer to semirings with additive inverses as “rigs”, see i.e. rig category, 2-rig, and division rig.

    • CommentRowNumber7.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 3rd 2022

    Ok, rigs are defined like this on the nlab. But I didn’t know it. It would have been more friendly to give more explanations…

    • CommentRowNumber8.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 3rd 2022

    6: you mean semirings with 0 are rigs on the nlab. I got it, it’s logical.

    • CommentRowNumber9.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    The term “semiring” has different definitions depending on the specific author. In Kazimierz Glazek’s A guide to the literature on semirings and their applications in mathematics and information sciences semirings come neither with 0 nor 1. The same is true of Wolfram MathWorld’s definition of semiring.

    • CommentRowNumber10.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 3rd 2022
    • (edited Aug 3rd 2022)

    Thank you for the explanation and the references.

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