nLab rig

Idea

In algebra, by a rig one means a mathematical structure much like a ring but without the assumption that every element has an additive inverse, hence without the assumption of negatives (whence the omission of “n” from “ring” [Schanuel 1991 p. 379, Lawvere 1992 p. 2])

Remark

(Terminology: Rigs and semirings) Rigs are commonly also called semirings, but the term ‘semiring’ is overloaded in the mathematics literature, with different authors each defining a semiring to be different algebraic structures from each other. See semiring for a discussion about the various definitions of semirings; only one of the proposed definitions is the same as the one of rigs as considered here.

Definition

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

• $R$ is a monoid under multiplication;
• $R$ is a commutative monoid under addition;
• multiplication distributes over addition, i.e. the distributivity laws hold:
  $$x\cdot (y+z) = (x\cdot y) + (x\cdot z)$$
  $$(y+z)\cdot x = (y\cdot x) + (z\cdot x)$$

  and also the absorption laws, which are the nullary version of the distributive laws:

  $$0\cdot x = 0 = x\cdot 0.$$

In a ring, the absorption laws follow from distributivity, since for example $0\cdot x + 0\cdot x = (0+0)\cdot x = 0\cdot x$ and we can cancel one copy to obtain $0\cdot x = 0$. In a rig, however, we have to assert the absorption laws 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 commutative monoids. Here the categories of abelian groups and commutative monoids must be given suitable monoidal structures: not the cartesian product, but the tensor product $\otimes$ that has a universal property for bilinear maps.

Equivalently, a rig is the hom-set of a category with a single object that is enriched in the category of commutative monoids.

Rigs and rig homomorphisms form the category Rig.

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 $R$ 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. 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, simple rig in which both operations are idempotent (see (Golan 2003, Proposition 2.25)). 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 generalization of how 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.

More formally, the ring completion of a rig $R$ is obtained by applying the group completion functor to the underlying additive monoid of $R$, and extending the rig multiplication to a ring multiplication by exploiting distributivity; this gives the left adjoint $F: Rig \to Ring$ to the forgetful functor $U: Ring \to Rig$. Note however that the unit of the adjunction $R \to U F(R)$ is not monic if the additive monoid of $R$ is not cancellative?, despite an informal convention that “completion” should usually mean a monad where the unit is monic.

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

Every rig with positive characteristic is a ring.

Examples

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 $\mathbb{R}\cup \{\infty\}$ with addition $x\oplus y = min(x,y)$ and multiplication $x\otimes y = x+y$.

  Tropical rigs are among the 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).

Related concepts

• semiring

• rng (nonunital ring)

• ring, near-ring

• tropical geometry, tropical semiring

• idempotent semiring

• A categorification of the notion of rig is the notion of rig category, or more generally colax-distributive rig category. See also 2-rig and distributivity for monoidal structures.

• semi-monoid/semi-group

• semi-category, semi-Segal space

• semi-simplicial set

• Burnside rig

• division rig

• multiplicatively cancellable rig

• quasiregular rig

• Conway rig

• Kleene star algebra

• additively idempotent rig

• multiplicatively idempotent rig

• Boolean rig (Guzmán)

• characteristic of a rig

References

The terminology “rig” is due to:

• Stephen H. Schanuel, p. 379 of: Negative sets have Euler characteristic and dimension, in: Category Theory, Lecture Notes in Mathematics 1488 (1991) 379–385 [doi:10.1007/BFb0084232]

• William Lawvere, pp. 1 of: Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf, pdf]

as recalled in:

• F. William Lawvere: The legacy of Steve Schanuel! (2015) [pdf, archive]

“We were amused when we finally revealed to each other that we had each independently come up with the term ‘rig’.”

Discussion under the name semirings:

• 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.

• Jonathan S. Golan, Semirings and affine equations over them: theory and applications (Vol. 556). Springer Science & Business Media, 2003.

• M. Marcolli, R. Thomgren, Thermodynamical semirings, arXiv/1108.2874

• wikipedia semiring

• J. Jun, S. Ray, J. Tolliver, Lattices, spectral spaces, and closure operations on idempotent semirings, arxiv/2001.00808

• Manfred Droste, Werner Kuich, Semirings and Formal Power Series (2009). In: Manfred Droste, Werner Kuich, Heiko Vogler (editors), Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. pp. 3–28.(doi:10.1007/978-3-642-01492-5_1)

• Zoltán Ésik, Iteration semirings (2008). In: Masami Ito, Masafumi Toyama (editors), Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings. Lecture Notes in Computer Science. Vol. 5257. Berlin: Springer-Verlag. pp. 1–20. (doi:10.1007/978-3-540-85780-8_1)

• Dexter Kozen (1990). On Kleene algebras and closed semirings. In: Branislav Rovan (editor). Mathematical Foundations of Computer Science 1990. MFCS 1990. Lecture Notes in Computer Science, vol 452. Springer, Berlin, Heidelberg. (doi:10.1007/BFb0029594, pdf)