Editing Semiring

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{{Short description|Algebraic ring that need not have additive negative elements}}
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In [[abstract algebra]], a '''semiring''' is an [[algebraic structure]]. Semirings are a generalization of [[Ring (algebra)|rings]], dropping the requirement that each element must have an [[additive inverse]]. At the same time, semirings are a generalization of [[Lattice (order)#Bounded lattice|bounded]] [[distributive lattice]]s.

The smallest semiring that is not a ring is the [[two-element Boolean algebra]], for instance with [[logical disjunction]] <math>\lor</math> as addition. A motivating example that is neither a ring nor a lattice is the set of [[natural number]]s <math>\N</math> (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the [[function composition]] of [[endomorphism]]s over any [[commutative monoid]].

{{Algebraic structures |Ring}}

== Terminology ==
Some authors define semirings without the requirement for there to be a <math>0</math> or <math>1</math>. This makes the analogy between '''ring''' and {{em|semiring}} on the one hand and {{em|[[Group (mathematics)|group]]}} and {{em|[[semigroup]]}} on the other hand work more smoothly. These authors often use '''rig''' for the concept defined here.{{sfnp|Głazek|2002|p=7|ps=}}{{efn|[http://www.proofwiki.org/wiki/Definition:Rig For an example see the definition of rig on Proofwiki.org]}} This originated as a joke, suggesting that rigs are ri''n''gs without ''n''egative elements. (Akin to using ''[[rng (algebra)|rng]]'' to mean a r''i''ng without a multiplicative ''i''dentity.)

The term '''dioid''' (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzmann in 1972 to denote a semiring.{{refn|{{cite book|last=Kuntzmann|first=J.|title=Théorie des réseaux (graphes)|language=fr|zbl=0239.05101|location=Paris|publisher=Dunod|year=1972 }}}} (It is alternatively sometimes used for [[naturally ordered semiring]]s{{refn|[http://marcpouly.ch/pdf/internal_100712.pdf Semirings for breakfast], slide 17}} but the term was also used for idempotent subgroups by [[François Baccelli|Baccelli]] et al. in 1992.{{refn|{{cite book|last1=Baccelli|first1=François Louis|last2=Olsder|first2=Geert Jan|last3=Quadrat|first3=Jean-Pierre|last4=Cohen|first4=Guy|title=Synchronization and linearity. An algebra for discrete event systems|zbl=0824.93003|series=Wiley Series on Probability and Mathematical Statistics|location=Chichester|publisher=Wiley|year=1992 }}}})

== Definition ==
A '''semiring''' is a [[Set (mathematics)|set]] <math>R</math> equipped with two [[binary operation]]s <math>+</math> and <math>\cdot,</math> called addition and multiplication, such that:{{sfnp|Berstel|Perrin|1985|loc=[{{Google books|plainurl=y|id=GHJHqezwwpcC|page=26|text=a semiring K is a set equipped with two operations}} p. 26]|ps=}}{{sfnp|Lothaire|2005|p=211|ps=}}{{sfnp|Sakarovitch|2009|pp=27–28|ps=}}

* <math>(R, +)</math> is a [[commutative]] [[monoid]] with an [[identity element]] called <math>0</math>:
** <math>(a + b) + c = a + (b + c)</math>
** <math>0 + a = a</math>
** <math>a + 0 = a</math>
** <math>a + b = b + a</math>
* <math>(R, \,\cdot\,)</math> is a monoid with an identity element called <math>1</math>:
** <math>(a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
** <math>1 \cdot a = a</math>
** <math>a \cdot 1 = a</math>
Further, the following axioms tie to both operations:
* Through multiplication, any element is left- and right-[[Annihilating element|annihilated]] by the additive identity:
** <math>0 \cdot a = 0</math>
** <math>a \cdot 0 = 0</math>
* Multiplication left- and right-[[Distributive law|distributes]] over addition:
** <math>a \cdot (b + c) = (a \cdot b) + (a \cdot c)</math>
** <math>(b + c) \cdot a = (b \cdot a) + (c \cdot a)</math>

=== Notation ===
The symbol <math>\cdot</math> is usually omitted from the notation; that is, <math>a \cdot b</math> is just written <math>ab.</math>

Similarly, an [[order of operations]] is conventional, in which <math>\cdot</math> is applied before <math>+</math>. That is, <math>a + b\cdot c</math> denotes <math>a + (b\cdot c)</math>.

For the purpose of disambiguation, one may write <math>0_R</math> or <math>1_R</math> to emphasize which structure the units at hand belong to.

If <math>x\in R</math> is an element of a semiring and <math>n\in{\mathbb N}</math>, then <math>n</math>-times repeated multiplication of <math>x</math> with itself is denoted <math>x^n</math>, and one similarly writes <math>x\,n:=x+x+\cdots+x</math> for the <math>n</math>-times repeated addition.

== Construction of new semirings ==
The [[zero ring]] with underlying set <math>\{0\}</math> is a semiring called the trivial semiring. This triviality can be characterized via <math>0=1</math> and so when speaking of nontrivial semirings, <math>0\neq 1</math> is often silently assumed as if it were an additional axiom.
Now given any semiring, there are several ways to define new ones.

As noted, the natural numbers <math>{\mathbb N}</math> with its arithmetic structure form a semiring. Taking the zero and the image of the successor operation in a semiring <math>R</math>, i.e., the set <math>\{x\in R\mid x=0_R\lor \exists p. x = p + 1_R\}</math> together with the inherited operations, is always a sub-semiring of <math>R</math>.

If <math>(M, +)</math> is a commutative monoid, function composition provides the multiplication to form a semiring: The set <math>\operatorname{End}(M)</math> of endomorphisms <math>M \to M</math> forms a semiring where addition is defined from pointwise addition in <math>M</math>. The [[zero morphism]] and the identity are the respective neutral elements. If <math>M = R^n</math> with <math>R</math> a semiring, we obtain a semiring that can be associated with the square <math>n\times n</math> [[Matrix (mathematics)|matrices]] <math>{\mathcal M}_n(R)</math> with coefficients in <math>R</math>, the [[matrix semiring]] using ordinary [[Matrix addition|addition]] and [[Matrix multiplication|multiplication]] rules of matrices. Given <math>n\in{\mathbb N}</math> and <math>R</math> a semiring, <math>{\mathcal M}_n(R)</math> is always a semiring also. It is generally non-commutative even if <math>R</math> was commutative.

[[Rng (algebra)#Adjoining an identity element (Dorroh extension)|Dorroh extensions]]: If <math>R</math> is a semiring, then <math>R\times{\mathbb N}</math> with pointwise addition and multiplication given by <math>\langle x,n\rangle\bullet \langle y,m\rangle:=\langle x\cdot y+(x\,m+y\,n), n\cdot m\rangle</math> defines another semiring with multiplicative unit <math>1_{R\times{\mathbb N}}:=\langle 0_R,1_{\mathbb N}\rangle</math>. Very similarly, if <math>N</math> is any sub-semiring of <math>R</math>, one may also define a semiring on <math>R\times N</math>, just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure <math>R</math> is not actually required to have a multiplicative unit.

[[Zerosumfree monoid|Zerosumfree]] semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero <math>0'</math> to the underlying set and thus obtain such a zerosumfree semiring that also lacks [[zero divisor]]s. In particular, now <math>0\cdot 0'=0'</math> and the old semiring is actually not a sub-semiring. One may then go on and adjoin new elements "on top" one at a time, while always respecting the zero. These two strategies also work under looser conditions. Sometimes the notations <math>-\infty</math> resp. <math>+\infty</math> are used when performing these constructions.

Adjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the [[logical connectives]] of disjunction and conjunction: <math>\langle\{0,1\},+,\cdot,\langle 0,1\rangle\rangle=\langle\{\bot,\top\},\lor,\land,\langle\bot,\top\rangle\rangle</math>. Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as <math>\top\lor P = \top</math> for all <math>P</math>, i.e. <math>1</math> has no additive inverse. In the [[Duality (order theory)|self-dual]] definition, the fault is with <math>\bot\land P = \bot</math>. (This is not to be conflated with the ring <math>\Z_2</math>, whose addition functions as [[xor]] <math>\veebar</math>.) 
In the [[Set-theoretic definition of natural numbers|von Neumann model of the naturals]], <math>0_\omega:=\{\}</math>, <math>1_\omega:=\{0_\omega\}</math> and <math>2_\omega:=\{0_\omega,1_\omega\}={\mathcal P}1_\omega</math>. The two-element semiring may be presented in terms of the set theoretic union and intersection as <math>\langle {\mathcal P}1_\omega,\cup,\cap,\langle \{\},1_\omega\rangle\rangle</math>. Now this structure in fact still constitutes a semiring when <math>1_\omega</math> is replaced by any inhabited set whatsoever.

The [[Ideal (ring theory)|ideals]] on a semiring <math>R</math>, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of <math>{\mathcal M}_n(R)</math> are in bijection with the ideals of <math>R</math>. The collection of left ideals of <math>R</math> (and likewise the right ideals) also have much of that algebraic structure, except that then <math>R</math> does not function as a two-sided multiplicative identity.

If <math>R</math> is a semiring and <math>A</math> is an [[inhabited set]], <math>A^*</math> denotes the [[free monoid]] and the formal polynomials <math>R[A^*]</math> over its words form another semiring. For small sets, the generating elements are conventionally used to denote the polynomial semiring. For example, in case of a singleton <math>A=\{X\}</math> such that <math>A^*=\{\varepsilon,X,X^2,X^3,\dots\}</math>, one writes <math>R[X]</math>. Zerosumfree sub-semirings of <math>R</math> can be used to determine sub-semirings of <math>R[A^*]</math>.

Given a set <math>A</math>, not necessarily just a singleton, adjoining a default element to the set underlying a semiring <math>R</math> one may define the semiring of partial functions from <math>A</math> to <math>R</math>.

Given a [[Derivation (differential algebra)|derivation]] <math>{\mathrm d}</math> on a semiring <math>R</math>, another the operation "<math>\bullet</math>" fulfilling <math>X\bullet y=y\bullet X+{\mathrm d}(y)</math> can be defined as part of a new multiplication on <math>R[X]</math>, resulting in another semiring.

The above is by no means an exhaustive list of systematic constructions.

=== Derivations ===
Derivations on a semiring <math>R</math> are the maps <math>{\mathrm d}\colon R\to R</math> with <math>{\mathrm d}(x+y)={\mathrm d}(x)+{\mathrm d}(y)</math> and <math>{\mathrm d}(x\cdot y)={\mathrm d}(x)\cdot y+x\cdot {\mathrm d}(y)</math>.

For example, if <math>E</math> is the <math>2\times 2</math> unit matrix and <math>U=\bigl(\begin{smallmatrix}0 & 1 \\ 0 & 0 \end{smallmatrix}\bigr)</math>, then the subset of <math>{\mathcal M}_2(R)</math> given by the matrices <math>a\,E + b\,U</math> with <math>a,b\in R</math> is a semiring with derivation <math>a\,E + b\,U\mapsto b\,U</math>.

== Properties ==
A basic property of semirings is that <math>1</math> is not a left or right [[zero divisor]], and that <math>1</math> but also <math>0</math> squares to itself, i.e. these have <math>u^2=u</math>.

Some notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty. Also, the [[Arity|2-ary]] predicate <math>x\le_\text{pre}y</math> defined as <math>\exists d. x + d = y</math>, here defined for the addition operation, always constitutes the right [[Monoid#Commutative monoid|canonical]] [[preorder]] relation. [[Reflexive relation|Reflexivity]] <math>y\le_\text{pre} y</math> is witnessed by the identity. Further, <math>0\le_\text{pre}y</math> is always valid, and so zero is the [[least element]] with respect to this preorder. Considering it for the commutative addition in particular, the distinction of "right" may be disregarded. In the non-negative integers <math>\N</math>, for example, this relation is [[anti-symmetric relation|anti-symmetric]] and [[strongly connected]], and thus in fact a (non-strict) [[total order]].

Below, more conditional properties are discussed.

=== Semifields ===
Any '''[[Field (mathematics)|field]]''' is also a '''[[semifield]]''', which in turn is a semiring in which also multiplicative inverses exist.

=== Rings ===
Any field is also a '''ring''', which in turn is a semiring in which also additive inverses exist. Note that a semiring omits such a requirement, i.e., it requires only a [[commutative monoid]], not a [[commutative group]]. The extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly.

Here <math>-1</math>, the additive inverse of <math>1</math>, squares to <math>1</math>. As additive differences <math>d=y-x</math> always exist in a ring, <math>x\le_\text{pre}y</math> is a trivial binary relation in a ring.

=== Commutative semirings ===
A semiring is called a '''[[commutative]] semiring''' if also the multiplication is commutative.{{sfnp|Lothaire|2005|p=212|ps=}} Its axioms can be stated concisely: It consists of two commutative monoids <math>\langle +, 0\rangle</math> and <math>\langle \cdot, 1\rangle</math> on one set such that <math>a\cdot 0 = 0</math> and <math>a\cdot (b+c)=a\cdot b + a\cdot c</math>.

The [[Center (ring theory)|center]] of a semiring is a sub-semiring and being commutative is equivalent to being its own center.

The commutative semiring of natural numbers is the [[initial object]] among its kind, meaning there is a unique structure preserving map of <math>{\mathbb N}</math> into any commutative semiring.

The bounded distributive lattices are [[partial order|partially ordered]], commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their [[Duality theory for distributive lattices|duals]].

=== Ordered semirings ===
Notions or order can be defined using strict, non-strict or [[second-order logic|second-order]] formulations. Additional properties such as commutativity simplify the axioms.

Given a [[strict total order]] (also sometimes called linear order, or [[pseudo-order]] in a constructive formulation), then by definition, the ''positive'' and ''negative'' elements fulfill <math>0<x</math> resp. <math>x<0</math>. By irreflexivity of a strict order, if <math>s</math> is a left zero divisor, then <math>s\cdot x < s\cdot y</math> is false. The ''non-negative'' elements are characterized by <math>\neg(x<0)</math>, which is then written <math>0\le x</math>.

Generally, the strict total order can be negated to define an associated partial order. The [[asymmetric relation|asymmetry]] of the former manifests as <math>x<y\to x\le y</math>. In fact in [[classical logic|classical mathematics]] the latter is a (non-strict) total order and such that <math>0\le x</math> implies <math>x=0\lor 0<x</math>. Likewise, given any (non-strict) total order, its negation is [[irreflexive relation|irreflexive]] and [[transitive relation|transitive]], and those two properties found together are sometimes called strict quasi-order. Classically this defines a strict total order – indeed strict total order and total order can there be defined in terms of one another.

Recall that "<math>\le_\text{pre}</math>" defined above is trivial in any ring. The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily coincide with "<math>\le_\text{pre}</math>".

==== Additively idempotent semirings ====
A semiring in which every element is an additive [[idempotent]], that is, <math>x+x=x</math> for all elements <math>x</math>, is called an '''(additively) idempotent semiring'''.{{refn|name=Esik08}} Establishing <math>1 + 1 = 1</math> suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication.

In such a semiring, <math>x\le_\text{pre} y</math> is equivalent to <math>x + y = y</math> and always constitutes a partial order, here now denoted <math>x\le y</math>. In particular, here <math>x \le 0\leftrightarrow x = 0</math>. So additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the trivial ring and so this property is specific to semiring theory. Addition and multiplication respect the ordering in the sense that <math>x \le y</math> implies <math>x + t \leq y + t</math>, and furthermore implies <math>s\cdot x \le s\cdot y</math> as well as <math>x\cdot s \le y\cdot s</math>, for all <math>x, y, t</math> and <math>s</math>.

If <math>R</math> is additively idempotent, then so are the polynomials in <math>R[X^*]</math>.

A semiring such that there is a lattice structure on its underlying set is '''lattice-ordered''' if the sum coincides with the meet, <math>x + y = x\lor y</math>, and the product lies beneath the join <math>x\cdot y \le x\land y</math>. The lattice-ordered semiring of ideals on a semiring is not necessarily [[distributive lattice|distributive with respect to]] the lattice structure.

More strictly than just additive idempotence, a semiring is called '''simple''' iff <math>x+1=1</math> for all <math>x</math>. Then also <math>1+1=1</math> and <math>x \le 1</math> for all <math>x</math>. Here <math>1</math> then functions akin to an additively infinite element. If <math>R</math> is an additively idempotent semiring, then <math>\{x\in R\mid x+1=1\}</math> with the inherited operations is its simple sub-semiring. 
An example of an additively idempotent semiring that is not simple is the [[tropical semiring]] on <math>{\mathbb R}\cup\{-\infty\}</math> with the 2-ary maximum function, with respect to the standard order, as addition. Its simple sub-semiring is trivial.

A '''c-semiring''' is an idempotent semiring and with addition defined over arbitrary sets.

An additively idempotent semiring with idempotent multiplication, <math>x^2=x</math>, is called '''additively and multiplicatively idempotent semiring''', but sometimes also just idempotent semiring. The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units). [[Heyting algebras]] are such semirings and the [[Boolean algebra (structure)|Boolean algebra]]s are a special case.

Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures.

==== Number lines ====
In a model of the ring <math>{\mathbb R}</math>, one can define a non-trivial positivity predicate <math>0<x</math> and a predicate <math>x<y</math> as <math>0<(y-x)</math> that constitutes a strict total order, which fulfills properties such as <math>\neg(x<0 \lor 0<x) \to x=0</math>, or classically the [[law of trichotomy]]. With its standard addition and multiplication, this structure forms the strictly [[ordered field]] that is [[Dedekind-complete]]. [[elementary equivalence|By definition]], all [[first-order logic|first-order properties]] proven in the theory of the reals are also provable in the [[Decidability (logic)#Decidability of a theory|decidable theory]] of the [[real closed field]]. For example, here <math> x < y </math> is mutually exclusive with <math>\exists d. y + d^2 = x</math>.

But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of <math>{\mathbb R}</math>, including the rationals, the integers, as well as the non-negative parts of each of these structures. In particular, the non-negative reals, the non-negative rationals and the non-negative integers are such a semirings.
The first two properties are analogous to the property valid in the idempotent semirings: Translation and scaling respect these [[ordered ring]]s, in the sense that addition and multiplication in this ring validate
* <math>(x<y)\,\to\,x+t<y+t</math>
* <math>(x<y\land 0<s)\,\to\,s\cdot x < s\cdot y</math>
In particular, <math>(0<y\land 0<s)\to 0 < s\cdot y</math> and so squaring of elements preserves positivity.

Take note of two more properties that are always valid in a ring. Firstly, trivially <math>P\,\to\,x \le_\text{pre} y</math> for any <math>P</math>. In particular, the ''positive'' additive difference existence can be expressed as
* <math>(x<y)\,\to\,x \le_\text{pre} y</math>
Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them. With <math>(-1)^2=1</math>, all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit,
* <math>0<1</math>
Having discussed a strict order, it follows that <math>0\neq 1</math> and <math>1\neq 1+1</math>, etc.

==== Discretely ordered semirings ====
There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by <math>1</math> being positive and [[Covering relation|covering]] <math>0</math>, i.e. there being no element <math>x</math> between the units, <math>\neg(0<x \land x<1)</math>. Now in the present context, an order shall be called '''discrete''' if this is fulfilled and, furthermore, all elements of the semiring are non-negative, so that the semiring starts out with the units.

Denote by <math>{\mathsf {PA}}^-</math> the theory of a commutative, discretely ordered semiring also validating the above four properties relating a strict order with the algebraic structure. All of its models have the model <math>\N</math> as its initial segment and [[Gödel's theorems|Gödel incompleteness]] and [[Tarski undefinability theorem|Tarski undefinability]] already apply to <math>{\mathsf {PA}}^-</math>. The non-negative elements of a commutative, [[Ordered ring#Discrete ordered rings|discretely ordered ring]] always validate the axioms of <math>{\mathsf {PA}}^-</math>. So a slightly more exotic model of the theory is given by the positive elements in the [[polynomial ring]] <math>{\mathbb Z}[X]</math>, with positivity predicate for <math> p={\textstyle\sum}_{k=0}^n a_k X^k </math> defined in terms of the last non-zero coefficient, <math>0 < p := (0 < a_n) </math>, and <math>p < q := (0 < q - p)</math> as above. While <math>{\mathsf {PA}}^-</math> proves all [[Arithmetical hierarchy|<math>\Sigma_1</math>-sentences]] that are true about <math>\N</math>, beyond this complexity one can find simple such statements that are [[logical independence|independent]] of <math>{\mathsf {PA}}^-</math>. For example, while <math>\Pi_1</math>-sentences true about <math>\N</math> are still true for the other model just defined, inspection of the polynomial <math>X</math> demonstrates <math>{\mathsf {PA}}^-</math>-independence of the <math>\Pi_2</math>-claim that all numbers are of the form <math>2q</math> or <math>2q+1</math> ("[[Parity (mathematics)#Definition|odd or even]]"). Showing that also <math>{\mathbb Z}[X,Y]/(X^2-2Y^2)</math> can be discretely ordered demonstrates that the <math>\Pi_1</math>-claim <math>x^2\neq 2y^2</math> for non-zero <math>x</math> ("no rational squared equals <math>2</math>") is independent. Likewise, analysis for <math>{\mathbb Z}[X,Y,Z]/(XZ-Y^2)</math> demonstrates independence of some statements about [[factorization]] true in <math>\N</math>. There are <math>{\mathsf {PA}}</math> characterizations of primality that <math>{\mathsf {PA}}^-</math> does not validate for the number <math>2</math>.

In the other direction, from any model of <math>{\mathsf {PA}}^-</math> one may construct an ordered ring, which then has elements that are negative with respect to the order, that is still discrete the sense that <math>1</math> covers <math>0</math>. To this end one defines an equivalence class of pairs from the original semiring. Roughly, the ring corresponds to the differences of elements in the old structure, generalizing the way in which the [[initial object|initial]] ring <math>\Z</math> [[Integer#Equivalence classes of ordered pairs|can be defined from]] <math>\N</math>. This, in effect, adds all the inverses and then the preorder is again trivial in that <math>\forall x. x\le_\text{pre} 0</math>.

Beyond the size of the two-element algebra, no simple semiring starts out with the units. Being discretely ordered also stands in contrast to, e.g., the standard ordering on the semiring of non-negative rationals <math>{\mathbb Q}_{\ge 0}</math>, which is [[dense order|dense]] between the units. For another example, <math>{\mathbb Z}[X]/(2X^2-1)</math> can be ordered, but not discretely so.

==== Natural numbers ====
<math>{\mathsf {PA}}^-</math> plus [[mathematical induction]] gives [[Peano axioms#Equivalent axiomatizations|a theory equivalent to]] first-order [[Peano arithmetic]] <math>{\mathsf {PA}}</math>. The theory is also famously not [[categorical theory|categorical]], but <math>\N</math> is of course the intended model. <math>{\mathsf {PA}}</math> proves that there are no zero divisors and it is zerosumfree and so no [[Non-standard model of arithmetic|model of it]] is a ring.

The standard axiomatization of <math>{\mathsf {PA}}</math> is more concise and the theory of its order is commonly treated in terms of the non-strict "<math>\le_\text{pre}</math>". However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory. Indeed, even [[Robinson arithmetic]] <math>{\mathsf {Q}}</math>, which removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom <math>\forall y. (0 + y = y)</math>.

=== Complete semirings ===
A '''complete semiring''' is a semiring for which the additive monoid is a [[complete monoid]], meaning that it has an [[Finitary|infinitary]] sum operation <math>\Sigma_I</math> for any [[index set]] <math>I</math> and that the following (infinitary) distributive laws must hold:<ref name=Kuich11/>{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}<ref>{{cite book|last=Kuich|first=Werner|chapter=ω-continuous semirings, algebraic systems and pushdown automata|pages=[https://archive.org/details/automatalanguage0000ical/page/103 103–110]|title=Automata, Languages and Programming: 17th International Colloquium, Warwick University, England, July 16–20, 1990, Proceedings|volume=443|series=Lecture Notes in Computer Science|editor1-first=Michael S.|editor1-last=Paterson|publisher=[[Springer-Verlag]]|year=1990|isbn=3-540-52826-1|chapter-url=https://archive.org/details/automatalanguage0000ical/page/103 }}</ref>
: <math>{\textstyle\sum}_{i \in I}{\left(a \cdot a_i\right)} = a \cdot \left({\textstyle\sum}_{i \in I}{a_i}\right), \qquad {\textstyle\sum}_{i \in I}{\left(a_i \cdot a\right)} = \left({\textstyle\sum}_{i \in I}{a_i}\right) \cdot a.</math>
Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.{{sfnp|Sakarovitch|2009|p=471|ps=}}
For commutative, additively idempotent and simple semirings, this property is related to [[residuated lattice]]s.

==== Continuous semirings ====
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#Generalization to ordered sets|least upper bound property]], and for which addition and multiplication respect order and suprema.  The semiring <math>\N \cup \{ \infty \}</math> with usual addition, multiplication and order extended is a continuous semiring.{{refn|{{cite book|last1=Ésik|first1=Zoltán|last2=Leiß|first2=Hans|chapter=Greibach normal form in algebraically complete semirings|zbl=1020.68056|editor1-last=Bradfield|editor1-first=Julian|title=Computer science logic. 16th international workshop, CSL 2002, 11th annual conference of the EACSL, Edinburgh, Scotland, September 22–25, 2002. Proceedings|location=Berlin|publisher=[[Springer-Verlag]]|series=Lecture Notes in Computer Science|volume=2471|pages=135–150|year=2002 }}}}

Any continuous semiring is complete:<ref name=Kuich11/> this may be taken as part of the definition.{{sfnp|Sakarovitch|2009|p=471|ps=}}

=== Star semirings ===
A '''star semiring''' (sometimes spelled '''starsemiring''') or '''[[closed semiring]]''' is a semiring with an additional unary operator <math>{}^*</math>,{{refn|name=Esik08}}{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}{{refn|{{citation|last1=Lehmann|first1=Daniel J.|title=Algebraic structures for transitive closure|journal=Theoretical Computer Science|volume=4|issue=1|year=1977|pages=59–76|doi=10.1016/0304-3975(77)90056-1 |url=http://wrap.warwick.ac.uk/46308/7/WRAP_Lehmann_cs-rr-010.pdf }}}}{{sfnp|Berstel|Reutenauer|2011|p=27|ps=}} satisfying
: <math>a^* = 1 + a a^* = 1 + a^* a.</math>
A '''[[Kleene algebra]]''' is a star semiring with idempotent addition and some additional axioms. They are important in the theory of [[formal language]]s and [[regular expression]]s.{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}

==== Complete star semirings ====
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:{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
: <math>a^* = {\textstyle\sum}_{j \geq 0}{a^j},</math>
where
: <math>a^j = \begin{cases}
  1,                                 & j = 0,\\
  a \cdot a^{j-1} = a^{j-1} \cdot a, & j > 0.
\end{cases}</math> 
Note that star semirings are not related to [[*-algebra]], where the star operation should instead be thought of as [[complex conjugation]].

==== Conway semiring ====

A '''Conway semiring''' is a star semiring satisfying the sum-star and product-star equations:{{refn|name=Esik08}}{{refn|
{{cite book|last1=Ésik|first1=Zoltán|last2=Kuich|first2=Werner|chapter=Equational axioms for a theory of automata|editor1-last=Martín-Vide|editor1-first=Carlos|title=Formal languages and applications|location=Berlin|publisher=[[Springer-Verlag]]|series=Studies in Fuzziness and Soft Computing|volume=148|pages=183–196|year=2004|isbn=3-540-20907-7|zbl=1088.68117}}}}
:<math>\begin{align}
  (a + b)^* &= \left(a^* b\right)^* a^*, \\
     (ab)^* &= 1 + a(ba)^* b.
\end{align}</math>

Every complete star semiring is also a Conway semiring,{{sfnp|Droste|Kuich|2009|p=15|loc=Theorem 3.4|ps=}} but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative [[rational number]]s <math>\Q_{\geq 0} \cup \{ \infty \}</math> 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).{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
An '''iteration semiring''' is a Conway semiring satisfying the Conway group axioms,{{refn|name=Esik08|{{cite book|last=Ésik|first=Zoltán|chapter=Iteration semirings|zbl=1161.68598|editor1-last=Ito|editor1-first=Masami|title=Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings|location=Berlin|publisher=[[Springer-Verlag]]|isbn=978-3-540-85779-2|series=Lecture Notes in Computer Science|volume=5257|pages=1–20|year=2008|doi=10.1007/978-3-540-85780-8_1}}}} associated by [[John Horton Conway|John Conway]] to groups in star-semirings.{{refn|{{cite book|first=J.H.|last=Conway|author-link=John Horton Conway|title=Regular algebra and finite machines|publisher=Chapman and Hall|year=1971|isbn=0-412-10620-5|zbl=0231.94041|location=London }}}}

== Examples ==
* By definition, any ring and any semifield is also a semiring.
* The non-negative elements of a commutative, discretely ordered ring form a commutative, discretely (in the sense defined above) ordered semiring. This includes the non-negative integers <math>\N</math>.
* Also the non-negative [[rational number]]s as well as the non-negative [[real number]]s form commutative, ordered semirings.<ref name=Gut08/>{{sfnp|Sakarovitch|2009|p=28}}{{sfnp|Berstel|Reutenauer|2011|p=4|ps=}} The latter is called ''{{visible anchor|probability semiring}}''.{{sfnp|Lothaire|2005|p=211|ps=}} Neither are rings or distributive lattices. These examples also have multiplicative inverses.
* New semirings can conditionally be constructed from existing ones, as described. The [[extended natural numbers]] <math>\N \cup \{ \infty \}</math> with addition and multiplication extended so that <math>0 \cdot \infty = 0</math>.{{sfnp|Sakarovitch|2009|p=28}}
* The set of [[polynomial]]s with natural number coefficients, denoted <math>\N[x],</math> forms a commutative semiring. In fact, this is the [[Free object|free]] commutative semiring on a single generator <math>\{ x \}.</math> Also polynomials with coefficients in other semirings may be defined, as discussed.
* The non-negative [[Positional notation#Terminating fractions|terminating fractions]] <math>\tfrac{\N}{b^{\N}} := \left\{ mb^{-n} \mid m, n \in \N \right\}</math>, in a [[Positional notation|positional number system]] to a given base <math>b\in \N</math>, form a sub-semiring of the rationals. One has <math>\tfrac{\N}{b^{\N}} \subseteq \tfrac{\N}{c^{\N}}</math>{{zwj}} if <math>b</math> divides <math>c</math>. For <math>|b| > 1</math>, the set <math>\tfrac{\Z_0}{b^{\Z_0}} := \tfrac{\N}{b^{\N}} \cup \left(-\tfrac{\N_0}{b^{\N}}\right) </math> is the ring of all terminating fractions to base <math>b,</math> and is [[Dense set|dense]] in <math>\Q</math>.
* The ''[[log semiring]]'' on <math>\R \cup \{ \pm \infty \}</math> with addition given by <math> x \oplus y = - \log\left(e^{-x} + e^{-y}\right)</math> with multiplication <math>+,</math> zero element <math>+ \infty,</math> and unit element <math>0.</math>{{sfnp|Lothaire|2005|p=211|ps=}}

Similarly, in the presence of an appropriate order with bottom element,
* [[Tropical semiring]]s are variously defined. The {{em|max-plus}} semiring <math>\R \cup \{ - \infty \}</math> is a commutative semiring with <math>\max(a, b)</math> serving as semiring addition (identity <math>- \infty</math>) and ordinary addition (identity 0) serving as semiring multiplication.  In an alternative formulation, the tropical semiring is <math>\R \cup \{ \infty \},</math> and min replaces max as the addition operation.{{refn|{{cite journal|last1=Speyer|first1=David|last2=Sturmfels|first2=Bernd|author2-link=Bernd Sturmfels|arxiv=math/0408099|title=Tropical Mathematics|orig-year=2004|year= 2009|zbl=1227.14051|journal=Math. Mag.|volume=82|number=3|pages=163–173|doi=10.4169/193009809x468760|s2cid=119142649 }}}}  A related version has <math>\R \cup \{ \pm \infty \}</math> as the underlying set.{{sfnp|Lothaire|2005|p=211|ps=}}<ref name=Kuich11>{{cite book|last=Kuich|first=Werner|chapter=Algebraic systems and pushdown automata|zbl=1251.68135|editor1-last=Kuich|editor1-first=Werner|title=Algebraic foundations in computer science. Essays dedicated to Symeon Bozapalidis on the occasion of his retirement|location=Berlin|publisher=[[Springer-Verlag]]|isbn=978-3-642-24896-2|series=Lecture Notes in Computer Science|volume=7020|pages=228–256|year=2011 }}</ref> They are an active area of research, linking [[Algebraic variety|algebraic varieties]] with [[Piecewise linear manifold|piecewise linear]] structures.{{refn|{{Cite journal|last1=Speyer|first1=David|last2=Sturmfels|first2=Bernd|date=2009|title=Tropical Mathematics|url=https://www.tandfonline.com/doi/full/10.1080/0025570X.2009.11953615|journal=Mathematics Magazine|language=en|volume=82|issue=3|pages=163–173|doi=10.1080/0025570X.2009.11953615|s2cid=15278805|issn=0025-570X}}}}
* The [[Jan Łukasiewicz|Łukasiewicz]] semiring: the closed interval <math>[0, 1]</math> with addition of <math>a</math> and <math>b</math> given by taking the maximum of the arguments (<math>\max(a, b)</math>) and multiplication of <math>a</math> and <math>b</math> given by <math>\max(0, a + b - 1)</math> appears in [[multi-valued logic]].{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
* The [[Andrew Viterbi|Viterbi]] semiring is also defined over the base set <math>[0, 1]</math> and has the maximum as its addition, but its multiplication is the usual multiplication of real numbers. It appears in [[probabilistic parsing]].{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}

* The set of all ideals of a given semiring form a semiring under addition and multiplication of ideals.
* Any bounded, distributive lattice is a commutative, semiring under join and meet. A Boolean algebra is a special case of these. A [[Boolean ring]] is also a semiring (indeed, a ring) but it is not idempotent under {{em|addition}}.  A {{em|Boolean semiring}} is a semiring isomorphic to a sub-semiring of a Boolean algebra.<ref name=Gut08>{{cite book|title=Surveys in Contemporary Mathematics|volume=347|series=London Mathematical Society Lecture Note Series|issn=0076-0552|editor1-first=Nicholas|editor1-last=Young|editor2-first=Yemon|editor2-last=Choi|publisher=[[Cambridge University Press]]|year=2008|isbn=978-0-521-70564-6|chapter=Rank and determinant functions for matrices over semirings|first=Alexander E.|last=Guterman|pages=1–33|zbl=1181.16042}}</ref>
* The commutative semiring formed by the two-element Boolean algebra and defined by <math>1 + 1 = 1</math>. It is also called ''the {{visible anchor|Boolean semiring}}''.{{sfnp|Lothaire|2005|p=211|ps=}}{{sfnp|Sakarovitch|2009|p=28}}{{sfnp|Berstel|Reutenauer|2011|p=4|ps=}}{{refn|name=Esik08}} Now given two sets <math>X</math> and <math>Y,</math> [[binary relation]]s between <math>X</math> and <math>Y</math> correspond to matrices indexed by <math>X</math> and <math>Y</math> with entries in the Boolean semiring, [[matrix addition]] corresponds to union of relations, and [[matrix multiplication]] corresponds to [[composition of relations]].{{refn|{{cite newsgroup|title=quantum mechanics over a commutative rig|author=John C. Baez|author-link=John C. Baez|date=6 Nov 2001|newsgroup=sci.physics.research|message-id=9s87n0$iv5@gap.cco.caltech.edu|url=https://groups.google.com/d/msg/sci.physics.research/VJNPMCfreao/TMKt9tFYNwEJ|access-date=November 25, 2018}}}}
* Any [[Quantale|unital quantale]] is a semiring under join and multiplication.
* A normal [[skew lattice]] in a ring <math>R</math> is a semiring for the operations multiplication and nabla, where the latter operation is defined by <math>a \nabla b = a + b + ba - aba - bab</math>

More using monoids,
* The construction of semirings <math>\operatorname{End}(M)</math> from a commutative monoid <math>M</math> has been described. As noted, give a semiring <math>R</math>, the <math>n\times n</math> matrices form another semiring. For example, the matrices with non-negative entries, <math>{{\mathcal M}}_n(\N),</math> form a matrix semiring.<ref name=Gut08/>
* {{anchor|formal languages}}Given an alphabet (finite set) Σ, the set of [[formal language]]s over <math>\Sigma</math> (subsets of [[Kleene star|<math>\Sigma^*</math>]]) is a semiring with product induced by [[string concatenation]] <math>L_1 \cdot L_2 = \left\{ w_1 w_2 \mid w_1 \in L_1, w_2 \in L_2 \right\}</math> and addition as the union of languages (that is, ordinary 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]].{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
* Generalizing the previous example (by viewing <math>\Sigma^*</math> as the [[free monoid]] over <math>\Sigma</math>), take <math>M</math> to be any monoid; the power set <math>\wp(M)</math> of all subsets of <math>M</math> forms a semiring under set-theoretic union as addition and set-wise multiplication: <math>U \cdot V = \{ u \cdot v \mid u \in U,\ v \in V \}.</math>{{sfnp|Berstel|Reutenauer|2011|p=4|ps=}}
* Similarly, if <math>(M, e, \cdot)</math> is a monoid, then the set of finite [[multiset]]s in <math>M</math> forms a semiring. That is, an element is a function <math>f \mid M \to \N</math>; given an element of <math>M,</math> 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 <math>e</math> to <math>1,</math> and all other elements of <math>M</math> to <math>0.</math> The sum is given by <math>(f + g)(x) = f(x) + g(x)</math> and the product is given by <math>(fg)(x) = \sum\{ f(y) g(z) \mid y \cdot z = x \}.</math>

Regarding sets and similar abstractions,
* {{anchor|binary relations}}Given a set <math>U,</math> the set of [[binary relation]]s over <math>U</math> 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]].{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}} These relations correspond to the [[matrix semiring]] (indeed, matrix semialgebra) of [[square matrices]] indexed by <math>U</math> 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 [[cardinal number]]s smaller than any given [[Infinity|infinite]] cardinal form a semiring under cardinal addition and multiplication. The class of {{em|all cardinals}} of an [[inner model]] form a (class) semiring under (inner model) cardinal addition and multiplication.
* The family of (isomorphism equivalence classes of) [[combinatorial class]]es (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.{{refn|{{citation|title=Algebraic Cryptanalysis|first=Gregory V.|last=Bard|publisher=Springer|year=2009|isbn=9780387887579|at=Section 4.2.1, "Combinatorial Classes", ff., pp. 30–34|url=https://books.google.com/books?id=kjbp0mgu3IAC&pg=PA30}}}}
* Isomorphism classes of objects in any [[distributive category]], under [[coproduct]] and [[Product (category theory)|product]] operations, form a semiring known as a Burnside rig.{{refn|Schanuel S.H. (1991) Negative sets have Euler characteristic and dimension. In: Carboni A., Pedicchio M.C., Rosolini G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg}} A Burnside rig is a ring if and only if the category is [[Category of small categories|trivial]].

=== Star semirings ===
Several structures mentioned above can be equipped with a star operation.
* The [[#binary relations|aforementioned]] semiring of [[binary relation]]s over some base set <math>U</math> in which <math>R^* = \bigcup_{n \geq 0} R^n</math> for all <math>R\subseteq U \times U.</math> This star operation is actually the [[Reflexive closure|reflexive]] and [[transitive closure]] of <math>R</math> (that is, the smallest reflexive and transitive binary relation over <math>U</math> containing <math>R.</math>).{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
* The [[#formal languages|semiring of formal languages]] is also a complete star semiring, with the star operation coinciding with the Kleene star (for sets/languages).{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
* The set of non-negative [[extended real]]s <math>[0, \infty]</math> together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by <math>a^* = \tfrac{1}{1 - a}</math> for <math>0 \leq a < 1</math> (that is, the [[geometric series]]) and <math>a^* = \infty</math> for <math>a \geq 1.</math>{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
* The Boolean semiring with <math>0^* = 1^* = 1.</math>{{efn|name=conway|This is a complete star semiring and thus also a Conway semiring.{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}}}{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
* The semiring on <math>\N \cup \{ \infty \},</math> with extended addition and multiplication, and <math>0^* = 1, a^* = \infty</math> for <math>a \geq 1.</math>{{efn|name=conway}}{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}

== Applications ==
The <math>(\max, +)</math> and <math>(\min, +)</math> [[tropical semiring]]s 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 path]]s can thus be reformulated as a computation over a <math>(\min, +)</math> 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 <math>(\max, \times)</math> 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.{{sfnp|Pair|1967|page=271 |s=}}{{sfnp|Derniame|Pair|1971|ps=}}

== Generalizations ==
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 {{em|hemirings}}{{sfnp|Golan|1999|p=1|loc=Ch 1|ps=}} or {{em|pre-semirings}}.{{sfnp|Gondran|Minoux|2008|p=22|loc=Ch 1, §4.2}} A further generalization are {{em|left-pre-semirings}},{{sfnp|Gondran|Minoux|2008|p=20|loc=Ch 1, §4.1}} which additionally do not require right-distributivity (or {{em|right-pre-semirings}}, which do not require left-distributivity).

Yet a further generalization are {{em|[[near-semiring]]s}}: 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 number]]s form a [[near-semiring]], when the standard [[Ordinal arithmetic|ordinal addition and multiplication]] are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called [[Ordinal arithmetic#Natural operations|natural (or Hessenberg) operations]] instead.

In [[category theory]], a {{em|2-rig}} is a category with [[functor]]ial 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 also ==
* {{annotated link|Ring of sets}}
* {{annotated link|Valuation algebra}}

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== Bibliography ==
{{refbegin}}
* {{citation |last1=Derniame |first1=Jean Claude |last2=Pair |first2=Claude |year=1971 |title=Problèmes de cheminement dans les graphes (Path Problems in Graphs) |publication-place=Paris| publisher=Dunod }}
* {{citation |last1=Baccelli |first1=François |author1-link=François Baccelli |first2=Guy |last2=Cohen |first3=Geert Jan |last3=Olsder |first4=Jean-Pierre |last4=Quadrat |url=http://cermics.enpc.fr/~cohen-g//SED/book-online.html |title=Synchronization and Linearity (online version) |publisher=Wiley |year=1992 |isbn=0-471-93609-X }}
* Golan, Jonathan S. (1999) ''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, {{MathSciNet|id=1163371}}). Kluwer Academic Publishers, Dordrecht. xii+381 pp. {{isbn|0-7923-5786-8}} {{MathSciNet|id=1746739}}
* {{cite book |last1=Berstel |first1=Jean |last2=Perrin |first2=Dominique |title=Theory of codes |series=Pure and applied mathematics|volume=117|year=1985|publisher=Academic Press|isbn=978-0-12-093420-1|zbl=0587.68066}}
* {{cite book |last1=Berstel |first1=Jean |last2=Reutenauer |first2=Christophe |title=Noncommutative rational series with applications |series=Encyclopedia of Mathematics and Its Applications |volume=137 |location=Cambridge |publisher=[[Cambridge University Press]] |year=2011 |isbn=978-0-521-19022-0 |zbl=1250.68007 }}
* {{citation |last1=Droste |first1=Manfred |last2=Kuich |first2=Werner |year=2009 |title=Handbook of Weighted Automata |chapter=Chapter 1: Semirings and Formal Power Series |pages=3–28 |doi=10.1007/978-3-642-01492-5_1}}
* {{Durrett Probability Theory and Examples 5th Edition}}
* {{citation |last1=Folland |first1=Gerald B. |year=1999 |title=Real Analysis: Modern Techniques and Their Applications |edition=2nd |publisher=John Wiley & Sons |isbn=9780471317166 |url=https://books.google.com/books?id=N8jVDwAAQBAJ&pg=PA23 }}
*{{citation
 | last = Golan | first = Jonathan S.
 | doi = 10.1007/978-94-015-9333-5
 | isbn = 0-7923-5786-8
 | location = Dordrecht
 | mr = 1746739
 | publisher = Kluwer Academic Publishers
 | title = Semirings and their Applications
 | year = 1999}}
* {{cite book |last1=Lothaire |first1=M. |author1-link=M. Lothaire |title=Applied combinatorics on words |others=A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, [[Gesine Reinert]], [[Sophie Schbath]], Michael Waterman, Philippe Jacquet, [[Wojciech Szpankowski]], Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and [[Valérie Berthé]] |series=Encyclopedia of Mathematics and Its Applications |volume=105 |location=Cambridge |publisher=[[Cambridge University Press]] |year=2005 |isbn=0-521-84802-4 |zbl=1133.68067 |url-access=registration |url=https://archive.org/details/appliedcombinato0000loth }}
* {{cite book |last1=Głazek |first1=Kazimierz |title=A guide to the literature on semirings and their applications in mathematics and information sciences. With complete bibliography |location=Dordrecht |publisher=Kluwer Academic |year=2002 |isbn=1-4020-0717-5 |zbl=1072.16040 }}
* {{cite book |last1=Gondran |first1=Michel |last2=Minoux |first2=Michel |year=2008 |title=Graphs, Dioids and Semirings: New Models and Algorithms |location=Dordrecht |publisher=Springer Science & Business Media |isbn=978-0-387-75450-5 |zbl=1201.16038 |series=Operations Research/Computer Science Interfaces Series |volume=41 }}
* {{citation |last1=Pair |first1=Claude |chapter=Sur des algorithmes pour des problèmes de cheminement dans les graphes finis (On algorithms for path problems in finite graphs) |title=Théorie des graphes (journées internationales d'études) – Theory of Graphs (international symposium) |publisher=Dunod (Paris) et Gordon and Breach (New York) |date=1967 |location=Rome (Italy), July 1966 |editor=Rosentiehl}}
* {{cite book |last=Sakarovitch |first=Jacques |title=Elements of automata theory |others=Translated from the French by Reuben Thomas |location=Cambridge |publisher=[[Cambridge University Press]] |year=2009 |isbn=978-0-521-84425-3|zbl=1188.68177 }}
{{refend}}

== Further reading ==
{{refbegin}}
* {{cite book |first1=Jonathan S. |last1=Golan |year=2003 |title=Semirings and Affine Equations over Them |publisher=Springer Science & Business Media |isbn=978-1-4020-1358-4 |zbl=1042.16038}}
* {{cite journal |last1=Grillet |first1=Mireille P. |title=Green's relations in a semiring|zbl=0227.16029|journal=Port. Math.|volume=29|pages=181–195|year=1970|url=https://eudml.org/doc/115127 }}
* {{cite book |last1=Gunawardena |first1=Jeremy |chapter=An introduction to idempotency |zbl=0898.16032 |editor1-last=Gunawardena |editor1-first=Jeremy |title=Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994 |location=Cambridge |publisher=[[Cambridge University Press]] |pages=1–49 |year=1998 |url=http://www.hpl.hp.com/techreports/96/HPL-BRIMS-96-24.pdf }}
* {{cite journal |last=Jipsen |first=P. |title=From semirings to residuated Kleene lattices|journal=Studia Logica|volume=76|number=2|year=2004|pages=291–303|zbl=1045.03049|doi=10.1023/B:STUD.0000032089.54776.63|s2cid=9946523 }}
* {{citation |last1=Dolan |first1=Steven |title=Proceedings of the 18th ACM SIGPLAN international conference on Functional programming |year=2013 |chapter-url=http://www.cl.cam.ac.uk/~sd601/papers/semirings.pdf |chapter=Fun with Semirings |pages=101–110 |doi=10.1145/2500365.2500613 |isbn=9781450323260 |s2cid=2436826 }}
{{refend}}

{{Authority control}}

[[Category:Algebraic structures]]
[[Category:Ring theory]]