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nLab > Latest Changes: multiplicatively idempotent semiring

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  1.  
    • CommentRowNumber1.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 19th 2022
    • (edited Dec 19th 2022)

    Added as examples: 0, ℤ/2ℤ and 𝔹={0,1} with 1+1=0. Proved that they are exactly the boolean rigs of cardinal less or equal than 2.

    I don’t know if boolean rigs in the sense of this entry are always commutative. In The variety of Boolean semirings, they show that they are commutative assuming that 1+x+x=1. If some noncommutative boolean rig exists, it must be of cardinal ≥3, not be a ring (because boolean rings are commutative) and not verify this equation.

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeMay 22nd 2023

    Added related concepts section

    diff, v9, current

    • CommentRowNumber3.
    • CommentAuthorP
    • CommentTime6 days ago

    these objects are called “multiplicatively idempotent rigs” in the literature, e.g.

    diff, v11, current

    • CommentRowNumber4.
    • CommentAuthorP
    • CommentTime6 days ago

    moving the entire examples section to Boolean rig (Guzmán) since all the examples are of commutative multiplicative idempotent rigs.

    diff, v11, current

    • CommentRowNumber5.
    • CommentAuthorP
    • CommentTime6 days ago

    Added fact that multiplicatively idempotent rigs are unital band objects in CMon

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorP
    • CommentTime5 days ago

    for that matter, the definition still works for semirings which are not rigs, so renaming this page to “multiplicatively idempotent semiring” in parallel to additively idempotent semiring

    diff, v13, current

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTime4 days ago

    Re #5 and #6: that sounds at least confusing, if not outright wrong. Normally, an “X object”, for some let’s say some algebraic notion X, would refer to an internalization of that notion in some brand of monoidal category that would support the expression of that notion. For example, the notion of monoid internalizes to any monoidal category, but the notion of commutative monoid would require an ambient symmetric or braided monoidal category. In the case of an idempotent monoid or band, the equation x⋅x=x involves a duplication of x on the left side, hence would require diagonal maps; the most natural scenario for that would be a cartesian monoidal category.

    But if you treat CMon as a cartesian monoidal category (in order to make sense of “idempotent monoid object”), then the internalization is not a multiplicative idempotent rig. I expect what you had in mind is rather (CMon,⊗), where monoid objects capture the distributivity (bilinearity of multiplication over addition), but then in that case, diagonals are problematic.

  8. removed incorrect statement

    Every multiplicatively idempotent semiring is an idempotent monoid object in CMon, the category of commutative monoids and monoid homomorphisms.

    Pete Sanders

    diff, v14, current

  9. Todd Trimble,

    On the other hand, there does exist the notion of monoidal category with diagonals, but I don’t if the tensor product of commutative monoids has diagonals in that sense.

    Edit: according to the Category Theory Zulip, the tensor product in both Ab and in CMon do not have diagonals in the above sense.

    • CommentRowNumber10.
    • CommentAuthorP
    • CommentTime4 days ago

    Pete Sanders is wrong. Idempotent monoids can be internalized in any monoidal category. See idempotent monoid in a monoidal category.

    diff, v15, current

  11. no they don’t, the “idempotent monoids” discussed in idempotent monoid in a monoidal category are a completely different concept compared to the idempotent monoids in algebra, as mentioned in the article itself.

    Pete Sanders

    diff, v16, current

1 to 11 of 11

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