nLab
rig

A rig is a ring ‘without negatives’ (hence the missing ‘n’ in the name, get it?). Similarly, a semiring has neither negatives nor even zero.

Specifically, it a set $R$ with operations of addition and multiplication, such that

• $R$ is a monoid under multiplication;
• $R$ is an abelian monoid (for a rig) or an abelian semigroup (for a semiring) under addition;
• multiplication distributes over addition.

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.

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.

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.

Some rigs which are neither rings nor distributive lattices include:

• The natural numbers.
• The nonnegative rational numbers and the nonnegative real numbers.
• Polynomials with coefficients in any rig.
• The set of isomorphism classes of objects in any distributive category, or more generally in any rig category?.
• The tropical rig?, which is $ℝ\cup \{\mathrm{\infty }\}$ with addition $x\oplus y=\min (x,y)$ and multiplication $x\otimes y=x+y$.
• 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).

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.