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nLab > Latest Changes: additively idempotent semiring

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  1. Added some remark on the order of a semiring. Actually, does anybody know if any semiring embedds into a semifield?

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 12th 2018

    Fixed the proof of the join property; added an observation about the example of languages.

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 12th 2018

    Seems a distributive lattice is a semiring; if a semifield is a semiring where multiplication by a non-zero xx is invertible, then x∧x=x∧⊤x \wedge x = x \wedge \top implies x=⊤x = \top assuming embeddability in a semifield. So, the answer is no.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 12th 2018

    Connes and Consani, for instance, say a semifield is a semiring (ie a rig) where the nonzero elements are all invertible.

  5. Corrected typo (“concatentation” for “concatenation”).

    Anonymous

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeSep 8th 2021

    Added two related concepts.

    diff, v11, current

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeSep 9th 2021

    Adjusting text and adding more examples.

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorP
    • CommentTimeJun 13th 2025

    renaming page to “additively idempotent semiring” to distinguish from the related notion of “multiplicatively idempotent semiring” for which “idempotent semiring” is also sometimes used in the literature.

    diff, v14, current

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