The symbol is usually omitted from the notation; that is, is just written Similarly, an order of operations is conventional, in which is applied before ; that is, is
Compared to a ring, a semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoid, not a commutative group. In a ring, the additive inverse requirement implies the existence of a multiplicative zero, so here it must be specified explicitly. If a semiring's multiplication is commutative, then it is called a commutative semiring.
There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here.[note 1]
A semiring in which every element is an additive idempotent (that is, for all elements ) is called an idempotent semiring. Idempotent semirings are specific to semiring theory since any idempotent semiring that is also a ring is in fact trivial.[note 2] One can define a partial order on an idempotent semiring by setting whenever (or, equivalently, if there exists an such that ). The least element with respect to this order is meaning that for all Addition and multiplication respect the ordering in the sense that implies and and
The and 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.
In particular, a Boolean algebra is such a semiring. A Boolean ring is also a semiring (indeed, a ring) but it is not idempotent under addition. A Boolean semiring is a semiring isomorphic to a subsemiring of a Boolean algebra.
A normal skew lattice in a ring is an idempotent semiring for the operations multiplication and nabla, where the latter operation is defined by
Any c-semiring is also a semiring, where addition is idempotent and defined over arbitrary sets.
A semiring (of sets) is a (non-empty) collection of subsets of such that
If (3) holds, then if and only if
If then there exists a finite number of mutually disjoint sets such that
Conditions (2) and (3) together with imply that Such semirings are used in measure theory. An example of a semiring of sets is the collection of half-open, half-closed real intervals
A semialgebra or elementary family is a collection of subsets of satisfying the semiring properties except with (3) replaced with:
If then there exists a finite number of mutually disjoint sets such that
This condition is stronger than (3), which can be seen as follows. If is a semialgebra and , then we can write for disjoint . Then:
and every since it is closed under intersection, and disjoint since they are contained in the disjoint 's. Moreover the condition is strictly stronger: any that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set ).
Given a semiring the matrix semiring of the square matrices form a semiring under ordinary addition and multiplication of matrices, and this semiring of matrices is generally non-commutative even though may be commutative. For example, the matrices with non-negative entries, form a matrix semiring.
If is a commutative monoid, the set of endomorphisms forms a semiring, where addition is pointwise addition and multiplication is function composition. The zero morphism and the identity are the respective neutral elements. If is the additive monoid of natural numbers we obtain the semiring of natural numbers as if with a semiring, we obtain (after associating each morphism to a matrix) the semiring of square matrices with coefficients in and if is a (commutative) group, then is a (not necessarily commutative) ring.
The set of polynomials with natural number coefficients, denoted forms a commutative semiring. In fact, this is the free commutative semiring on a single generator
Tropical semirings are variously defined. The max-plus semiring is a commutative, idempotent semiring with serving as semiring addition (identity ) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is and min replaces max as the addition operation. A related version has as the underlying set.
The set of cardinal numbers smaller than any given infinite cardinal form a semiring under cardinal addition and multiplication. The class of all cardinals of an inner model form a (class) semiring under (inner model) cardinal addition and multiplication.
The probability semiring of non-negative real numbers under the usual addition and multiplication.
with multiplication zero element and unit element 
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.
The Łukasiewicz semiring: the closed interval with addition given by taking the maximum of the arguments () and multiplication given by appears in multi-valued logic.
The Viterbi semiring is also defined over the base set and has the maximum as its addition, but its multiplication is the usual multiplication of real numbers. It appears in probabilistic parsing.
Given an alphabet (finite set) Σ, the set of formal languages over (subsets of ) is a semiring with product induced by string concatenation 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.
Generalizing the previous example (by viewing as the free monoid over ), take to be any monoid; the power set of all subsets of forms a semiring under set-theoretic union as addition and set-wise multiplication: 
Similarly, if is a monoid, then the set of finite multisets in forms a semiring. That is, an element is a function ; given an element of the function tells you how many times that element occurs in the multiset it represents. The additive unit is the constant zero function. The multiplicative unit is the function mapping to and all other elements of to The sum is given by and the product is given by
Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.
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 with usual addition, multiplication and order extended is a continuous semiring.
Any continuous semiring is complete: this may be taken as part of the definition.
A star semiring (sometimes spelled starsemiring) is a semiring with an additional unary operator ∗, satisfying
A Conway semiring is a star semiring satisfying the sum-star and product-star equations:
Every complete star semiring is also a Conway semiring, but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers 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).
An iteration semiring is a Conway semiring satisfying the Conway group axioms, associated by John Conway to groups in star-semirings.
The set of non-negative extended reals together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by for (that is, the geometric series) and for 
A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called hemirings or pre-semirings. A further generalization are left-pre-semirings, which additionally do not require right-distributivity (or right-pre-semirings, which do not require left-distributivity).
In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.
Ring of sets – Family closed under unions and relative complements
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