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

Definition

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.

Examples

• The simplest nontrivial example is the rig N of natural numbers (including zero), with the ordinary addition and multiplication. Likewise, the non-negative rational numbers and the non-negative real numbers form rigs. All these rigs are commutative.
• Any ring is automatically also a rig.
• The ideals of a ring form a rig under addition and multiplication of ideals.
• Square n-by-n matrices with non-negative entries form a rig under ordinary addition and multiplication of matrices.
• Any distributive lattice is a rig under join and meet.
• In particular, a Boolean algebra is a rig under these operations (as well as a Boolean ring under different operations).
• The set of cardinal numbers smaller than any given infinite cardinal form a rig under cardinal addition and multiplication. (We can't form a rig of all cardinal numbers because they do not form a set.)

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.