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nLab > Latest Changes: 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

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