Main Page | See live article | Alphabetical index

Rig (algebra)

In abstract algebra, a rig is an algebraic structure, similar to a ring, but without an analogue of subtraction. The term "rig", which originated as a joke, is meant to suggest that rigs do not have "negative" elements.

Table of contents
1 Definition
2 Examples
3 Rig theory
4 Further generalisations


A rig is a set R equipped with two binary operations + and *, such that:

  1. + is a commutative monoid with identity element 0; that is:
    1. (a + b) + c = a + (b + c);
    2. 0 + a = a;
    3. a + 0 = a;
    4. a + b = b + a.
  2. * is a monoid with identity element 1; that is:
    1. (a * b) * c = a * (b * c);
    2. 1 * a = a;
    3. a * 1 = a.
  3. * distributes over +; that is:
    1. a * (b + c) = (a * b) + (a * c)
    2. (a + b) * c = (a * c) + (b * c)

The symbol * is usually omitted from the notation; that is, a * b is just written ab. Similarly, an order of operations is accepted, according to which * is applied before +; that is, a + bc is a + (b * c).

The difference between rings and rigs, then, is that the operation + yields only a monoid, not necessarily a group.

A rig is called commutative if its multiplication is commutative.


Rig theory

Much of the theory of rings continues to make sense when applied to arbitrary rigs. In particular, one can generalise the theory of algebras over commutative rings directly to a theory of algebras over commutative rigs. Then a ring is simply an algebra over the commutative rig Z of integers. Some mathematicians go so far as to say that rigs are really the more fundamental concept, and specialising to rings should be seen in the same light as specialising to, say, algebras over the complex numbers.

Further generalisations

A near-rig does not require addition to be commutative, nor does it require right-distributivity. That is, laws 1.4 and 3.2 in the definition above are dropped. Just as cardinal numbers form a rig, so do ordinal numbers form a near-rig.

In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.