Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 2nd 2022

    Added link to “multiplicatively cancellable semi-ring”.

    diff, v32, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    Semirings as defined on Wolfram MathWorld don’t have either an additive or multiplicative identity; they are semigroup objects in the category of commutative semigroups.

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022
    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 3rd 2022
    • (edited Aug 3rd 2022)

    Thank you, it seems very logical to me now. I will use the term rig for the structure with the two identities from now on.

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    I’ve seen four different definitions of a “semiring” out there, depending on the author:

    • A semigroup object in the category of commutative semigroups
    • A monoid object in the category of commutative semigroups
    • A semigroup object in the category of commutative monoids
    • A monoid object in the category of commutative monoids

    The problem is already there in the definition of a ring, as some authors define a ring to be a semigroup object in the category of abelian groups, while other authors define a ring to be a monoid object in the category of commutative monoids

    • CommentRowNumber6.
    • CommentAuthorGuest
    • CommentTimeAug 3rd 2022

    *define a ring to be a monoid object in the category of abelian groups

  1. adding paragraph on the relationship between rigs and semirings.


    diff, v35, current

    • CommentRowNumber8.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 18th 2022

    Added a reference to Rig

    diff, v38, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2023

    added pointer to

    • William Lawvere, pp. 1 of: Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf]

    Is this (on p. 2) maybe the actual origin of the term “rig”?

    diff, v40, current

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 25th 2023

    It appears earlier than 1992, e.g., in

    • S.H. Schanuel, Negative sets have Euler characteristic and dimension, in: Proceedings of Category Theory, Como, Italy,1990, in: Lecture Notes in Mathematics, vol. 1488, Springer-Verlag, 1991, pp. 379–385.

    See here, page 379. But I wouldn’t bet this is the earliest appearance either.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2023

    Thanks, I have added that to the entry.

    diff, v41, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023

    Thanks to a reference provided by Rod McGuire in another thread (here):

    we can settle the question of origin of the terminology ’rig’ – because Lawvere writes there, about his work with Schanuel, that:

    We were amused when we finally revealed to each other that we had each independently come up with the term ’rig’.

    Have added this to the entry.

    diff, v42, current