symmetric monoidal (∞,1)-category of spectra
A rig is a ring ‘without negatives’ (hence the missing ‘n’ in the name, get it?). Rigs are commonly also called semirings, but by analogy with semigroup it would be more appropriate to use that word for a ring having neither negatives nor even zero, so that is what we will do here.
A rig is a set $R$ with binary operations of addition and multiplication, such that
and also the absorption/annihilation laws, which are their nullary version:
In a ring, absorption follows from distributivity, since $0\cdot x + 0\cdot x = (0+0)\cdot x = 0\cdot x$ and we can cancel one copy to obtain $0\cdot x = 0$. In a rig, however, we have to assert absorption separately.
More sophisticatedly, we can say that, just as a ring is a monoid object in abelian groups, so a rig is a monoid object in abelian monoids and a semiring is a monoid object in abelian semigroups, where abelian groups, abelian monoids and abelian semigroups have suitable monoidal structures (they are not the cartesian ones).
Equivalently, a semiring is the hom-set of a category with a single object that is enriched in the category of abelian semigroups.
As with rings, one sometimes considers non-associative or non-unital versions (where multiplication may not be associative or may have no identity). It is rarer to remove requirements from addition as we have done here. But notice that while $R$ can be proved (from the other axioms) to be an abelian group under addition (and therefore a ring) as long as it is a group, this argument does not go through if it is only a monoid. If we assert only distributivity on one side, however, then we can have a noncommutative addition; see near-ring.
Many rigs are either rings or distributive lattices. Indeed, a ring is precisely a rig that forms a group under addition, while a distributive lattice is precisely a commutative, simple rig in which both operations are idempotent (see (Golan 2003, Proposition 2.25)). Note that a Boolean algebra is a rig in both ways: as a lattice and as a Boolean ring.
Any rig can be “completed” to a ring by adding negatives, in the same way that the natural numbers are completed to the integers. When applied to the set of isomorphism classes of objects in a rig category, the result is part of algebraic K-theory. More formally, the ring completion of a rig $R$ is obtained by applying the group completion functor to the underlying additive monoid of $R$, and extending the rig multiplication to a ring multiplication by exploiting distributivity; this gives the left adjoint $F: Rig \to Ring$ to the forgetful functor $U: Ring \to Rig$. Note however that the unit of the adjunction $R \to U F(R)$ is not monic if the additive monoid of $R$ is not cancellative?, despite an informal convention that “completion” should usually mean a monad where the unit is monic.
Matrices of rigs can be used to formulate versions of matrix mechanics.
Some rigs which are neither rings nor distributive lattices include:
Tropical rigs are one of an important class of idempotent semirings.
A categorification of the notion of rig is the notion of rig category, or more generally colax-distributive rig category. See also 2-rig and distributivity for monoidal structures.
Last revised on May 20, 2021 at 11:49:43. See the history of this page for a list of all contributions to it.