nLab
rig

Rigs and semirings

Idea

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.

Definition

A rig is a set RR with binary operations of addition and multiplication, such that

  • RR is a monoid under multiplication;
  • RR is an abelian monoid (for a rig) or an abelian semigroup (for a semiring) under addition;
  • multiplication distributes over addition, i.e. the distributivity laws hold:
    x(y+z)=(xy)+(xz)x\cdot (y+z) = (x\cdot y) + (x\cdot z)
    (y+z)x=(yx)+(zx)(y+z)\cdot x = (y\cdot x) + (z\cdot x)

    and also the absorption/annihilation laws, which are their nullary version:

    0x=0=x00\cdot x = 0 = x\cdot 0

In a ring, absorption follows from distributivity, since 0x+0x=(0+0)x=0x0\cdot x + 0\cdot x = (0+0)\cdot x = 0\cdot x and we can cancel one copy to obtain 0x=00\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.

Equivalently, a semiring is the hom-set of of a semicategory with a single object that is enriched in Ab.

Further weakening

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 RR 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.

Properties

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 rig in which the operations are idempotent. 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.

Matrices of rigs can be used to formulate versions of matrix mechanics.

Examples

Some rigs which are neither rings nor distributive lattices include:

Tropical rigs are one of an important class of idempotent semirings.

  • The ideals of a commutative ring form a rig under ideal addition and multiplication, where the unit and zero ideals are the unit and zero elements of the rig, respectively. They also form a distributive lattice and therefore a rig in another way; note that the addition operation is the same in both rigs but the multiplication operation is different (being intersection in the lattice).

References

  • Jonathan S. Golan, Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science, Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp.
  • M. Marcolli, R. Thomgren, Thermodynamical semirings, arXiv: 1108.2874
  • wikipedia semiring

Revised on July 17, 2013 09:11:17 by Mike Shulman (107.201.30.59)