semiring

Some authors define semirings without the requirement for there to be a $0$ or $1$. This makes the analogy between ring and {{em|semiring}} on the one hand and {{em|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 rings without negative elements. (Akin to using rng to mean a ring without a multiplicative identity.)

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 semirings{{refn|[http://marcpouly.ch/pdf/internal_100712.pdf Semirings for breakfast], slide 17}} but the term was also used for idempotent subgroups by 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 }}}})

A semiring is a set $R$ equipped with two binary operations $+$ and $\backslash cdot,$ 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=}}

• $(R,\; +)$ is a commutative monoid with an identity element called $0$:
• $(a\; +\; b)\; +\; c\; =\; a\; +\; (b\; +\; c)$
• $0\; +\; a\; =\; a$
• $a\; +\; 0\; =\; a$
• $a\; +\; b\; =\; b\; +\; a$
• $(R,\; \backslash ,\backslash cdot\backslash ,)$ is a monoid with an identity element called $1$:
• $(a\; \backslash cdot\; b)\; \backslash cdot\; c\; =\; a\; \backslash cdot\; (b\; \backslash cdot\; c)$
• $1\; \backslash cdot\; a\; =\; a$
• $a\; \backslash cdot\; 1\; =\; a$

Further, the following axioms tie to both operations:

• Through multiplication, any element is left- and right-annihilated by the additive identity:
• $0\; \backslash cdot\; a\; =\; 0$
• $a\; \backslash cdot\; 0\; =\; 0$
• Multiplication left- and right-distributes over addition:
• $a\; \backslash cdot\; (b\; +\; c)\; =\; (a\; \backslash cdot\; b)\; +\; (a\; \backslash cdot\; c)$
• $(b\; +\; c)\; \backslash cdot\; a\; =\; (b\; \backslash cdot\; a)\; +\; (c\; \backslash cdot\; a)$

The symbol $\backslash cdot$ is usually omitted from the notation; that is, $a\; \backslash cdot\; b$ is just written $ab.$

Similarly, an order of operations is conventional, in which $\backslash cdot$ is applied before $+$. That is, $a\; +\; b\backslash cdot\; c$ denotes $a\; +\; (b\backslash cdot\; c)$.

For the purpose of disambiguation, one may write $0\_R$ or $1\_R$ to emphasize which structure the units at hand belong to.

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

The zero ring with underlying set $\backslash \{0\backslash \}$ is a semiring called the trivial semiring. This triviality can be characterized via $0=1$ and so when speaking of nontrivial semirings, $0\backslash neq\; 1$ 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 $\{\backslash mathbb\; N\}$ with its arithmetic structure form a semiring. Taking the zero and the image of the successor operation in a semiring $R$, i.e., the set $\backslash \{x\backslash in\; R\backslash mid\; x=0\_R\backslash lor\; \backslash exists\; p.\; x\; =\; p\; +\; 1\_R\backslash \}$ together with the inherited operations, is always a sub-semiring of $R$.

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

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

Zerosumfree semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero $0\text{'}$ to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular, now $0\backslash cdot\; 0\text{'}=0\text{'}$ 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 $-\backslash infty$ resp. $+\backslash infty$ 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: $\backslash langle\backslash \{0,1\backslash \},+,\backslash cdot,\backslash langle\; 0,1\backslash rangle\backslash rangle=\backslash langle\backslash \{\backslash bot,\backslash top\backslash \},\backslash lor,\backslash land,\backslash langle\backslash bot,\backslash top\backslash rangle\backslash rangle$. Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as $\backslash top\backslash lor\; P\; =\; \backslash top$ for all $P$, i.e. $1$ has no additive inverse. In the self-dual definition, the fault is with $\backslash bot\backslash land\; P\; =\; \backslash bot$. (This is not to be conflated with the ring $\backslash Z\_2$, whose addition functions as xor $\backslash veebar$.)

In the von Neumann model of the naturals, $0\_\backslash omega:=\backslash \{\backslash \}$, $1\_\backslash omega:=\backslash \{0\_\backslash omega\backslash \}$ and $2\_\backslash omega:=\backslash \{0\_\backslash omega,1\_\backslash omega\backslash \}=\{\backslash mathcal\; P\}1\_\backslash omega$. The two-element semiring may be presented in terms of the set theoretic union and intersection as $\backslash langle\; \{\backslash mathcal\; P\}1\_\backslash omega,\backslash cup,\backslash cap,\backslash langle\; \backslash \{\backslash \},1\_\backslash omega\backslash rangle\backslash rangle$. Now this structure in fact still constitutes a semiring when $1\_\backslash omega$ is replaced by any inhabited set whatsoever.

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

If $R$ is a semiring and $A$ is an inhabited set, $A^*$ denotes the free monoid and the formal polynomials $R[A^*]$ 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 $A=\backslash \{X\backslash \}$ such that $A^*=\backslash \{\backslash varepsilon,X,X^2,X^3,\backslash dots\backslash \}$, one writes $R[X]$. Zerosumfree sub-semirings of $R$ can be used to determine sub-semirings of $R[A^*]$.

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

Given a derivation $\{\backslash mathrm\; d\}$ on a semiring $R$, another the operation "$\backslash bullet$" fulfilling $X\backslash bullet\; y=y\backslash bullet\; X+\{\backslash mathrm\; d\}(y)$ can be defined as part of a new multiplication on $R[X]$, resulting in another semiring.

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

Derivations on a semiring $R$ are the maps $\{\backslash mathrm\; d\}\backslash colon\; R\backslash to\; R$ with $\{\backslash mathrm\; d\}(x+y)=\{\backslash mathrm\; d\}(x)+\{\backslash mathrm\; d\}(y)$ and $\{\backslash mathrm\; d\}(x\backslash cdot\; y)=\{\backslash mathrm\; d\}(x)\backslash cdot\; y+x\backslash cdot\; \{\backslash mathrm\; d\}(y)$.

For example, if $E$ is the $2\backslash times\; 2$ unit matrix and , then the subset of $\{\backslash mathcal\; M\}\_2(R)$ given by the matrices $a\backslash ,E\; +\; b\backslash ,U$ with $a,b\backslash in\; R$ is a semiring with derivation $a\backslash ,E\; +\; b\backslash ,U\backslash mapsto\; b\backslash ,U$.

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

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 2-ary predicate $x\backslash le\_\backslash text\{pre\}y$ defined as $\backslash exists\; d.\; x\; +\; d\; =\; y$, here defined for the addition operation, always constitutes the right canonical preorder relation. Reflexivity $y\backslash le\_\backslash text\{pre\}\; y$ is witnessed by the identity. Further, $0\backslash le\_\backslash text\{pre\}y$ 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 $\backslash N$, for example, this relation is anti-symmetric and strongly connected, and thus in fact a (non-strict) total order.

Below, more conditional properties are discussed.

Any field is also a semifield, which in turn is a semiring in which also multiplicative inverses exist.

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 $-1$, the additive inverse of $1$, squares to $1$. As additive differences $d=y-x$ always exist in a ring, $x\backslash le\_\backslash text\{pre\}y$ is a trivial binary relation in a ring.

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 $\backslash langle\; +,\; 0\backslash rangle$ and $\backslash langle\; \backslash cdot,\; 1\backslash rangle$ on one set such that $a\backslash cdot\; 0\; =\; 0$ and $a\backslash cdot\; (b+c)=a\backslash cdot\; b\; +\; a\backslash cdot\; c$.

The 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 $\{\backslash mathbb\; N\}$ into any commutative semiring.

The bounded distributive lattices are partially ordered, commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their duals.

Notions or order can be defined using strict, non-strict or 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 non-negative elements are characterized by , which is then written $0\backslash le\; x$.

Generally, the strict total order can be negated to define an associated partial order. The asymmetry of the former manifests as

Recall that "$\backslash le\_\backslash text\{pre\}$" 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 "$\backslash le\_\backslash text\{pre\}$".

A semiring in which every element is an additive idempotent, that is, $x+x=x$ for all elements $x$, is called an (additively) idempotent semiring.{{refn|name=Esik08}} Establishing $1\; +\; 1\; =\; 1$ suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication.

In such a semiring, $x\backslash le\_\backslash text\{pre\}\; y$ is equivalent to $x\; +\; y\; =\; y$ and always constitutes a partial order, here now denoted $x\backslash le\; y$. In particular, here $x\; \backslash le\; 0\backslash leftrightarrow\; x\; =\; 0$. 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 $x\; \backslash le\; y$ implies $x\; +\; t\; \backslash leq\; y\; +\; t$, and furthermore implies $s\backslash cdot\; x\; \backslash le\; s\backslash cdot\; y$ as well as $x\backslash cdot\; s\; \backslash le\; y\backslash cdot\; s$, for all $x,\; y,\; t$ and $s$.

If $R$ is additively idempotent, then so are the polynomials in $R[X^*]$.

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

More strictly than just additive idempotence, a semiring is called simple iff $x+1=1$ for all $x$. Then also $1+1=1$ and $x\; \backslash le\; 1$ for all $x$. Here $1$ then functions akin to an additively infinite element. If $R$ is an additively idempotent semiring, then $\backslash \{x\backslash in\; R\backslash mid\; x+1=1\backslash \}$ 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 $\{\backslash mathbb\; R\}\backslash cup\backslash \{-\backslash infty\backslash \}$ 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, $x^2=x$, 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 algebras 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.

In a model of the ring $\{\backslash mathbb\; R\}$, one can define a non-trivial positivity predicate

But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of $\{\backslash mathbb\; R\}$, 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 rings, in the sense that addition and multiplication in this ring validate

In particular,

Take note of two more properties that are always valid in a ring. Firstly, trivially $P\backslash ,\backslash to\backslash ,x\; \backslash le\_\backslash text\{pre\}\; y$ for any $P$. In particular, the positive additive difference existence can be expressed as

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 $(-1)^2=1$, all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit,

Having discussed a strict order, it follows that $0\backslash neq\; 1$ and $1\backslash neq\; 1+1$, etc.

There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by $1$ being positive and covering $0$, i.e. there being no element $x$ between the units, 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 $\{\backslash mathsf\; \{PA\}\}^-$ 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 $\backslash N$ as its initial segment and Gödel incompleteness and Tarski undefinability already apply to $\{\backslash mathsf\; \{PA\}\}^-$. The non-negative elements of a commutative, discretely ordered ring always validate the axioms of $\{\backslash mathsf\; \{PA\}\}^-$. So a slightly more exotic model of the theory is given by the positive elements in the polynomial ring $\{\backslash mathbb\; Z\}[X]$, with positivity predicate for $p=\{\backslash textstyle\backslash sum\}\_\{k=0\}^n\; a\_k\; X^k$ defined in terms of the last non-zero coefficient, , and as above. While $\{\backslash mathsf\; \{PA\}\}^-$ proves all $\backslash Sigma\_1$-sentences that are true about $\backslash N$, beyond this complexity one can find simple such statements that are independent of $\{\backslash mathsf\; \{PA\}\}^-$. For example, while $\backslash Pi\_1$-sentences true about $\backslash N$ are still true for the other model just defined, inspection of the polynomial $X$ demonstrates $\{\backslash mathsf\; \{PA\}\}^-$-independence of the $\backslash Pi\_2$-claim that all numbers are of the form $2q$ or $2q+1$ ("odd or even"). Showing that also $\{\backslash mathbb\; Z\}[X,Y]/(X^2-2Y^2)$ can be discretely ordered demonstrates that the $\backslash Pi\_1$-claim $x^2\backslash neq\; 2y^2$ for non-zero $x$ ("no rational squared equals $2$") is independent. Likewise, analysis for $\{\backslash mathbb\; Z\}[X,Y,Z]/(XZ-Y^2)$ demonstrates independence of some statements about factorization true in $\backslash N$. There are $\{\backslash mathsf\; \{PA\}\}$ characterizations of primality that $\{\backslash mathsf\; \{PA\}\}^-$ does not validate for the number $2$.

In the other direction, from any model of $\{\backslash mathsf\; \{PA\}\}^-$ 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 $1$ covers $0$. 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 ring $\backslash Z$ can be defined from $\backslash N$. This, in effect, adds all the inverses and then the preorder is again trivial in that $\backslash forall\; x.\; x\backslash le\_\backslash text\{pre\}\; 0$.

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 $\{\backslash mathbb\; Q\}\_\{\backslash ge\; 0\}$, which is dense between the units. For another example, $\{\backslash mathbb\; Z\}[X]/(2X^2-1)$ can be ordered, but not discretely so.

$\{\backslash mathsf\; \{PA\}\}^-$ plus mathematical induction gives a theory equivalent to first-order Peano arithmetic $\{\backslash mathsf\; \{PA\}\}$. The theory is also famously not categorical, but $\backslash N$ is of course the intended model. $\{\backslash mathsf\; \{PA\}\}$ proves that there are no zero divisors and it is zerosumfree and so no model of it is a ring.

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

A complete semiring is a semiring for which the additive monoid is a complete monoid, meaning that it has an infinitary sum operation $\backslash Sigma\_I$ for any index set $I$ and that the following (infinitary) distributive laws must hold:{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}{{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 }}

: $\{\backslash textstyle\backslash sum\}\_\{i\; \backslash in\; I\}\{\backslash left(a\; \backslash cdot\; a\_i\backslash right)\}\; =\; a\; \backslash cdot\; \backslash left(\{\backslash textstyle\backslash sum\}\_\{i\; \backslash in\; I\}\{a\_i\}\backslash right),\; \backslash qquad\; \{\backslash textstyle\backslash sum\}\_\{i\; \backslash in\; I\}\{\backslash left(a\_i\; \backslash cdot\; a\backslash right)\}\; =\; \backslash left(\{\backslash textstyle\backslash sum\}\_\{i\; \backslash in\; I\}\{a\_i\}\backslash right)\; \backslash cdot\; a.$

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 lattices.

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 $\backslash N\; \backslash cup\; \backslash \{\; \backslash infty\; \backslash \}$ 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: this may be taken as part of the definition.{{sfnp|Sakarovitch|2009|p=471|ps=}}

A star semiring (sometimes spelled starsemiring) or closed semiring is a semiring with an additional unary operator $\{\}^*$,{{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

: $a^*\; =\; 1\; +\; a\; a^*\; =\; 1\; +\; a^*\; a.$

A Kleene algebra is a star semiring with idempotent addition and some additional axioms. They are important in the theory of formal languages and regular expressions.{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}

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=}}

: $a^*\; =\; \{\backslash textstyle\backslash sum\}\_\{j\; \backslash geq\; 0\}\{a^j\},$

where

: $a^j\; =\; \backslash begin\{cases\}$

1, & j = 0,\\

a \cdot a^{j-1} = a^{j-1} \cdot a, & j > 0.

\end{cases}

Note that star semirings are not related to *-algebra, where the star operation should instead be thought of as complex conjugation.

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}}}}

:$\backslash begin\{align\}$

(a + b)^* &= \left(a^* b\right)^* a^*, \\

(ab)^* &= 1 + a(ba)^* b.

\end{align}

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 numbers $\backslash Q\_\{\backslash geq\; 0\}\; \backslash cup\; \backslash \{\; \backslash infty\; \backslash \}$ 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 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 }}}}

• 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 $\backslash N$.
• Also the non-negative rational numbers as well as the non-negative real numbers form commutative, ordered semirings.{{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 $\backslash N\; \backslash cup\; \backslash \{\; \backslash infty\; \backslash \}$ with addition and multiplication extended so that $0\; \backslash cdot\; \backslash infty\; =\; 0$.{{sfnp|Sakarovitch|2009|p=28}}
• The set of polynomials with natural number coefficients, denoted $\backslash N[x],$ forms a commutative semiring. In fact, this is the free commutative semiring on a single generator $\backslash \{\; x\; \backslash \}.$ Also polynomials with coefficients in other semirings may be defined, as discussed.
• The non-negative terminating fractions $\backslash tfrac\{\backslash N\}\{b^\{\backslash N\}\}\; :=\; \backslash left\backslash \{\; mb^\{-n\}\; \backslash mid\; m,\; n\; \backslash in\; \backslash N\; \backslash right\backslash \}$, in a positional number system to a given base $b\backslash in\; \backslash N$, form a sub-semiring of the rationals. One has $\backslash tfrac\{\backslash N\}\{b^\{\backslash N\}\}\; \backslash subseteq\; \backslash tfrac\{\backslash N\}\{c^\{\backslash N\}\}${{zwj}} if $b$ divides $c$. For $|b|\; >\; 1$, the set $\backslash tfrac\{\backslash Z\_0\}\{b^\{\backslash Z\_0\}\}\; :=\; \backslash tfrac\{\backslash N\}\{b^\{\backslash N\}\}\; \backslash cup\; \backslash left(-\backslash tfrac\{\backslash N\_0\}\{b^\{\backslash N\}\}\backslash right)$ is the ring of all terminating fractions to base $b,$ and is dense in $\backslash Q$.
• The log semiring on $\backslash R\; \backslash cup\; \backslash \{\; \backslash pm\; \backslash infty\; \backslash \}$ with addition given by $x\; \backslash oplus\; y\; =\; -\; \backslash log\backslash left(e^\{-x\}\; +\; e^\{-y\}\backslash right)$ with multiplication $+,$ zero element $+\; \backslash infty,$ and unit element $0.${{sfnp|Lothaire|2005|p=211|ps=}}

Similarly, in the presence of an appropriate order with bottom element,

• Tropical semirings are variously defined. The {{em|max-plus}} semiring $\backslash R\; \backslash cup\; \backslash \{\; -\; \backslash infty\; \backslash \}$ is a commutative semiring with $\backslash max(a,\; b)$ serving as semiring addition (identity $-\; \backslash infty$) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is $\backslash R\; \backslash cup\; \backslash \{\; \backslash infty\; \backslash \},$ 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 $\backslash R\; \backslash cup\; \backslash \{\; \backslash pm\; \backslash infty\; \backslash \}$ as the underlying set.{{sfnp|Lothaire|2005|p=211|ps=}}{{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 }} They are an active area of research, linking algebraic varieties with 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 Łukasiewicz semiring: the closed interval $[0,\; 1]$ with addition of $a$ and $b$ given by taking the maximum of the arguments ($\backslash max(a,\; b)$) and multiplication of $a$ and $b$ given by $\backslash max(0,\; a\; +\; b\; -\; 1)$ appears in multi-valued logic.{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
• 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.{{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.{{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}}
• The commutative semiring formed by the two-element Boolean algebra and defined by $1\; +\; 1\; =\; 1$. 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 $X$ 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.{{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 unital quantale is a semiring under join and multiplication.
• A normal skew lattice in a ring $R$ is a semiring for the operations multiplication and nabla, where the latter operation is defined by $a\; \backslash nabla\; b\; =\; a\; +\; b\; +\; ba\; -\; aba\; -\; bab$

More using monoids,

• The construction of semirings $\backslash operatorname\{End\}(M)$ from a commutative monoid $M$ has been described. As noted, give a semiring $R$, the $n\backslash times\; n$ matrices form another semiring. For example, the matrices with non-negative entries, $\{\{\backslash mathcal\; M\}\}\_n(\backslash N),$ form a matrix semiring.
• {{anchor|formal languages}}Given an alphabet (finite set) Σ, the set of formal languages over $\backslash Sigma$ (subsets of $\backslash Sigma^*$) is a semiring with product induced by string concatenation $L\_1\; \backslash cdot\; L\_2\; =\; \backslash left\backslash \{\; w\_1\; w\_2\; \backslash mid\; w\_1\; \backslash in\; L\_1,\; w\_2\; \backslash in\; L\_2\; \backslash right\backslash \}$ 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 $\backslash Sigma^*$ as the free monoid over $\backslash Sigma$), take $M$ to be any monoid; the power set $\backslash wp(M)$ of all subsets of $M$ forms a semiring under set-theoretic union as addition and set-wise multiplication: $U\; \backslash cdot\; V\; =\; \backslash \{\; u\; \backslash cdot\; v\; \backslash mid\; u\; \backslash in\; U,\backslash \; v\; \backslash in\; V\; \backslash \}.${{sfnp|Berstel|Reutenauer|2011|p=4|ps=}}
• Similarly, if $(M,\; e,\; \backslash cdot)$ is a monoid, then the set of finite multisets in $M$ forms a semiring. That is, an element is a function $f\; \backslash mid\; M\; \backslash to\; \backslash N$; given an element of $M,$ 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 $e$ to $1,$ and all other elements of $M$ to $0.$ The sum is given by $(f\; +\; g)(x)\; =\; f(x)\; +\; g(x)$ and the product is given by $(fg)(x)\; =\; \backslash sum\backslash \{\; f(y)\; g(z)\; \backslash mid\; y\; \backslash cdot\; z\; =\; x\; \backslash \}.$

Regarding sets and similar abstractions,

• {{anchor|binary relations}}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.{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}} 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 cardinal numbers smaller than any given 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 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.{{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 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 trivial.

Several structures mentioned above can be equipped with a star operation.

• The aforementioned semiring of binary relations over some base set $U$ in which $R^*\; =\; \backslash bigcup\_\{n\; \backslash geq\; 0\}\; R^n$ for all $R\backslash subseteq\; U\; \backslash times\; U.$ This star operation is actually the reflexive and transitive closure of $R$ (that is, the smallest reflexive and transitive binary relation over $U$ containing $R.$).{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
• The 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 reals $[0,\; \backslash infty]$ together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by $a^*\; =\; \backslash tfrac\{1\}\{1\; -\; a\}$ for (that is, the geometric series) and $a^*\; =\; \backslash infty$ for $a\; \backslash geq\; 1.${{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}
• The Boolean semiring with $0^*\; =\; 1^*\; =\; 1.${{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 $\backslash N\; \backslash cup\; \backslash \{\; \backslash infty\; \backslash \},$ with extended addition and multiplication, and $0^*\; =\; 1,\; a^*\; =\; \backslash infty$ for $a\; \backslash geq\; 1.${{efn|name=conway}}{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}

The $(\backslash max,\; +)$ and $(\backslash min,\; +)$ 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 $(\backslash min,\; +)$ 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 $(\backslash max,\; \backslash times)$ 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=}}

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-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-semiring, 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 {{em|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.

• {{annotated link|Ring of sets}}
• {{annotated link|Valuation algebra}}

{{notelist}}

{{reflist}}

{{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}}

{{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 }}

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Category:Algebraic structures

Category:Ring theory