# Semiring

In abstract algebra, a **semiring** is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

The term **rig** is also used occasionally^{[1]}—this originated as a joke, suggesting that rigs are ri*n*gs without *n*egative elements, similar to using *rng* to mean a r*i*ng without a multiplicative *i*dentity.

Tropical semirings are an active area of research, linking algebraic varieties with piecewise linear structures.^{[2]}

## DefinitionEdit

A **semiring** is a set *R* equipped with two binary operations + and ⋅, called addition and multiplication, such that:^{[3]}^{[4]}^{[5]}

- (
*R*, +) is a commutative monoid with identity element 0:- (
*a*+*b*) +*c*=*a*+ (*b*+*c*) - 0 +
*a*=*a*+ 0 =*a* *a*+*b*=*b*+*a*

- (
- (
*R*, ⋅) is a monoid with identity element 1:- (
*a*⋅*b*)⋅*c*=*a*⋅(*b*⋅*c*) - 1⋅
*a*=*a*⋅1 =*a*

- (
- Multiplication left and right distributes over addition:
*a*⋅(*b*+*c*) = (*a*⋅*b*) + (*a*⋅*c*)- (
*a*+*b*)⋅*c*= (*a*⋅*c*) + (*b*⋅*c*)

- Multiplication by 0 annihilates
*R*:- 0⋅
*a*=*a*⋅0 = 0

- 0⋅

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* + (*bc*).

Compared to a ring, a semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoid, not a commutative group. In a ring, the additive inverse requirement implies the existence of a multiplicative zero, so here it must be specified explicitly. If a semiring's multiplication is commutative, then it is called a **commutative semiring**.^{[6]}

There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between *ring* and *semiring* on the one hand and *group* and *semigroup* on the other hand work more smoothly. These authors often use *rig* for the concept defined here.^{[note 1]}

## TheoryEdit

Much of the theory of rings continues to make sense when applied to arbitrary semirings^{[citation needed]}.
In particular, one can generalise the theory of (associative) algebras over commutative rings directly to a theory of algebras over commutative semirings.
Then a ring is simply an algebra over the commutative semiring **Z** of integers.

A semiring in which every element is an additive idempotent (that is, *a* + *a* = *a* for all elements *a*) is called an **idempotent semiring**.^{[7]} Idempotent semirings are special to semiring theory as any ring which is idempotent under addition is trivial.^{[note 2]} One can define a partial order ≤ on an idempotent semiring by setting *a* ≤ *b* whenever *a* + *b* = *b* (or, equivalently, if there exists an *x* such that *a* + *x* = *b*). It is easy to see that 0 is the least element with respect to this order: 0 ≤ *a* for all *a*. Addition and multiplication respect the ordering in the sense that *a* ≤ *b* implies *ac* ≤ *bc* and *ca* ≤ *cb* and (*a* + *c*) ≤ (*b* + *c*).

### ApplicationsEdit

The (max, +) and (min, +) tropical semirings on the reals, are often used in performance evaluation on discrete event systems. The real numbers then are the "costs" or "arrival time"; the "max" operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the "min" operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path.

The Floyd–Warshall algorithm for shortest paths can thus be reformulated as a computation over a (min, +) algebra. Similarly, the Viterbi algorithm for finding the most probable state sequence corresponding to an observation sequence in a Hidden Markov model can also be formulated as a computation over a (max, ×) algebra on probabilities. These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.^{[8]}^{[9]}

## ExamplesEdit

By definition, any ring is also a semiring. A motivating example of a semiring is the set of natural numbers **N** (including zero) under ordinary addition and multiplication. Likewise, the non-negative rational numbers and the non-negative real numbers form semirings. All these semirings are commutative.^{[10]}^{[11]}^{[12]}

### In generalEdit

- The set of all ideals of a given ring form an idempotent semiring under addition and multiplication of ideals.
- Any unital quantale is an idempotent semiring under join and multiplication.
- Any bounded, distributive lattice is a commutative, idempotent semiring under join and meet.
- In particular, a Boolean algebra is such a semiring. A Boolean ring is also a semiring (indeed, a ring) but it is not idempotent under
*addition*. A*Boolean semiring*is a semiring isomorphic to a subsemiring of a Boolean algebra.^{[10]} - A normal skew lattice in a ring
*R*is an idempotent semiring for the operations multiplication and nabla, where the latter operation is defined by . - Any c-semiring is also a semiring, where addition is idempotent and defined over arbitrary sets.
- Isomorphism classes of objects in any distributive category, under coproduct and product operations, form a semiring known as a Burnside rig.
^{[13]}A Burnside rig is a ring if and only if the category is trivial.

#### Semiring of setsEdit

A **semiring** (**of sets**)^{[14]} is a non-empty collection S of sets such that

- If and then .
- If and then there exists a finite number of mutually disjoint sets for such that .

Such semirings are used in measure theory. An example of a semiring of sets is the collection of half-open, half-closed real intervals .

### Specific examplesEdit

- The (non-negative)
*terminating fractions*in a positional number system to a given base . We have if divides . Furthermore, is the ring of all terminating fractions to base , and is dense in for . - The extended natural numbers
**N**∪ {∞} with addition and multiplication extended (and 0⋅∞ = 0).^{[11]} - Given a semiring
*S*, the matrix semiring of the square*n*-by-*n*matrices form a semiring under ordinary addition and multiplication of matrices, and this semiring of matrices is generally non-commutative even though*S*may be commutative. For example, the matrices with non-negative entries, , form a matrix semiring.^{[10]} - If
*A*is a commutative monoid, the set End(*A*) of endomorphisms*f*:*A*→*A*almost forms a semiring, where addition is pointwise addition and multiplication is function composition. The zero morphism and the identity are the respective neutral elements. This is not a true semiring because composition does not distribute on the left over pointwise addition:*a*· (*b*+*c*) ≠ (*a*·*b*) + (*a*·*c*). If*A*is the additive monoid of natural numbers we obtain the semiring of natural numbers as End(*A*), and if*A*=*S*^{n}with*S*a semiring, we obtain (after associating each morphism to a matrix) the semiring of square*n*-by-*n*matrices with coefficients in*S*. - The
**Boolean semiring**is the commutative semiring**B**formed by the two-element Boolean algebra and defined by 1 + 1 = 1.^{[4]}^{[11]}^{[12]}It is idempotent^{[7]}and is the simplest example of a semiring that is not a ring. Given two sets*X*and*Y*, binary relations between*X*and*Y*correspond to matrices indexed by*X*and*Y*with entries in the Boolean semiring, matrix addition corresponds to union of relations, and matrix multiplication corresponds to composition of relations.^{[15]} - Given a set
*U*, the set of binary relations over*U*is a semiring with addition the union (of relations as sets) and multiplication the composition of relations. The semiring's zero is the empty relation and its unit is the identity relation.^{[16]}These relations correspond to the matrix semiring (indeed, matrix semialgebra) of square matrices indexed by*U*with entries in the Boolean semiring, and then addition and multiplication are the usual matrix operations, while zero and the unit are the usual zero matrix and identity matrix. - The set of polynomials with natural number coefficients, denoted
**N**[*x*], forms a commutative semiring. In fact, this is the free commutative semiring on a single generator {*x*}. - Tropical semirings are variously defined. The
*max-plus*semiring**R**∪ {−∞} is a commutative, idempotent semiring with max(*a*,*b*) serving as semiring addition (identity −∞) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is**R**∪ {∞}, and min replaces max as the addition operation.^{[17]}A related version has**R**∪ {±∞} as the underlying set.^{[4]}^{[18]} - The set of cardinal numbers smaller than any given infinite cardinal form a semiring under cardinal addition and multiplication. The class of
*all cardinals*of an inner model form a (class) semiring under (inner model) cardinal addition and multiplication. - The
**probability semiring**of non-negative real numbers under the usual addition and multiplication.^{[4]} - The
**log semiring**on**R**∪ {±∞} with addition given by^{[4]} - The family of (isomorphism equivalence classes of) combinatorial classes (sets of countably many objects with non-negative integer sizes such that there are finitely many objects of each size) with the empty class as the zero object, the class consisting only of the empty set as the unit, disjoint union of classes as addition, and Cartesian product of classes as multiplication.
^{[19]} - The Łukasiewicz semiring: the closed interval [0, 1] with addition given by taking the maximum of the arguments (
*a*+*b*= max(*a*,*b*)) and multiplication*ab*given by max(0,*a*+*b*− 1) appears in multi-valued logic.^{[16]} - The Viterbi semiring is also defined over the base set [0, 1] and has the maximum as its addition, but its multiplication is the usual multiplication of real numbers. It appears in probabilistic parsing.
^{[16]} - Given an alphabet (finite set) Σ, the set of formal languages over Σ (subsets of Σ
^{∗}) is a semiring with product induced by string concatenation and addition as the union of languages (i.e. simply union as sets). The zero of this semiring is the empty set (empty language) and the semiring's unit is the language containing only the empty string.^{[16]} - Generalising the previous example (by viewing Σ
^{∗}as the free monoid over Σ), take*M*to be any monoid; the power set**P**(*M*) of all subsets of*M*forms a semiring under set-theoretic union as addition and set-wise multiplication: .^{[12]} - Similarly, if is a monoid, then the set of finite multisets in forms a semiring. That is, an element is a function ; given an element of , the function tells you how many times that element occurs in the multiset it represents. The additive unit is the constant zero function. The multiplicative unit is the function mapping to 1, and all other elements of to 0. The sum is given by and the product is given by .

## VariationsEdit

### Complete and continuous semiringsEdit

A **complete semiring** is a semiring for which the additive monoid is a complete monoid, meaning that it has an infinitary sum operation Σ_{I} for any index set *I* and that the following (infinitary) distributive laws must hold:^{[18]}^{[16]}^{[20]}

An examples of a complete semiring is the power set of a monoid under union. The matrix semiring over a complete semiring is complete.^{[21]}

A **continuous semiring** is similarly defined as one for which the addition monoid is a continuous monoid. That is, partially ordered with the least upper bound property, and for which addition and multiplication respect order and suprema. The semiring **N** ∪ {∞} with usual addition, multiplication and order extended is a continuous semiring.^{[22]}

Any continuous semiring is complete:^{[18]} this may be taken as part of the definition.^{[21]}

### Star semiringsEdit

A **star semiring** (sometimes spelled **starsemiring**) is a semiring with an additional unary operator ^{∗},^{[7]}^{[16]}^{[23]}^{[24]} satisfying

A **Kleene algebra** is a star semiring with idempotent addition. They are important in the theory of formal languages and regular expressions.^{[16]}

#### Complete star semiringsEdit

In a **complete star semiring**, the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:^{[16]}

where

#### Conway semiringEdit

A **Conway semiring** is a star semiring satisfying the sum-star and product-star equations:^{[7]}^{[25]}

Every complete star semiring is also a Conway semiring,^{[26]} but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers **Q**_{≥0} ∪ {∞} with the usual addition and multiplication (this is a modification of the example with extended non-negative reals given in this section by eliminating irrational numbers).^{[16]}

An **iteration semiring** is a Conway semiring satisfying the Conway group axioms,^{[7]} associated by John Conway to groups in star-semirings.^{[27]}

#### ExamplesEdit

Examples of star semirings include:

- the (aforementioned) semiring of binary relations over some base set
*U*in which for all . This star operation is actually the reflexive and transitive closure of*R*(i.e. the smallest reflexive and transitive binary relation over*U*containing*R*.).^{[16]} - the semiring of formal languages is also a complete star semiring, with the star operation coinciding with the Kleene star (for sets/languages).
^{[16]} - The set of non-negative extended reals [0, ∞] together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by
*a*^{∗}= 1/(1 −*a*) for 0 ≤*a*< 1 (i.e. the geometric series) and*a*^{∗}= ∞ for*a*≥ 1.^{[16]} - The Boolean semiring with 0
^{∗}= 1^{∗}= 1.^{[a]}^{[16]} - The semiring on
**N**∪ {∞}, with extended addition and multiplication, and 0^{∗}= 1,*a*^{∗}= ∞ for*a*≥ 1.^{[a]}^{[16]}

### DioidEdit

The term **dioid** (for "double monoid") has been used to mean various types of semirings:

- It was used by Kuntzman in 1972 to denote what is now termed semiring.
^{[28]} - The use to mean idempotent subgroup was introduced by Baccelli et al. in 1992.
^{[29]} - The name "dioid" is also sometimes used to denote naturally ordered semirings.
^{[30]}

## GeneralizationsEdit

A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called *hemirings*^{[31]} or *pre-semirings*.^{[32]} A further generalization are *left-pre-semirings*,^{[33]} which additionally do not require right-distributivity (or *right-pre-semirings*, which do not require left-distributivity).

Yet a further generalization are *near-semirings*: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-ring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead.

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.

## See alsoEdit

## NotesEdit

**^**For an example see the definition of rig on Proofwiki.org**^**i.e. is a ring consisting of just one element.

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## Further readingEdit

- Golan, Jonathan S. (2003).
*Semirings and Affine Equations over Them*. Springer Science & Business Media. ISBN 978-1-4020-1358-4. Zbl 1042.16038. - Gondran, Michel; Minoux, Michel (2008).
*Graphs, Dioids and Semirings: New Models and Algorithms*. Operations Research/Computer Science Interfaces Series.**41**. Dordrecht: Springer Science & Business Media. ISBN 978-0-387-75450-5. Zbl 1201.16038. - Grillet, Mireille P. (1970). "Green's relations in a semiring".
*Port. Math*.**29**: 181–195. Zbl 0227.16029. - Gunawardena, Jeremy (1998). "An introduction to idempotency". In Gunawardena, Jeremy (ed.).
*Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994*(PDF). Cambridge: Cambridge University Press. pp. 1–49. Zbl 0898.16032. - Jipsen, P. (2004). "From semirings to residuated Kleene lattices".
*Studia Logica*.**76**(2): 291–303. doi:10.1023/B:STUD.0000032089.54776.63. Zbl 1045.03049. - Steven Dolan (2013) Fun with Semirings, doi:10.1145/2500365.2500613