
A rig is a set R equipped with two binary operations + and *, such that:
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.
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.
A nearrig[?] does not require addition to be commutative, nor does it require rightdistributivity. 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 nearrig.
In category theory, a 2rig[?] 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 2rig.
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