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nLab > Latest Changes: division rig

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  1. division rigs, the rig version of division rings

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorAli Lahijani
    • CommentTimeJul 5th 2021
    Definition 1.3 refers to subtraction, but it's not clear how (if) it can be defined for a rig.
    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 5th 2021

    I guess one could apply the definition in the case that for every pair of elements x,yx,y there is a unique zz such that either x=y+zx=y+z or y=x+zy=x+z. Then it makes sense to define x#yx\# y iff such zz is invertible. This assumption should probably hold if the map to the ring completion is injective, which seems to require that the additive monoid is cancellative.

    One might demand that addition is cancellative in a general division rig, and I guess this might be enough to arrive at the definition otherwise involving subtraction. Or else add this as an explicit hypothesis to this definition, assuming it works.

  4. Corrected typo (missing plural on the word “rational”).

    Anonymous

    diff, v2, current

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