{"history":[],"article":{"id":"87c2d298-00dc-4d8a-9f02-a00d8853d0ba","title":"Semiring","slug":"semiring","content":":::summary\n:::\n\n# \nIn [abstract algebra](https://bliptext.com/articles/abstract-algebra), a **semiring** is an [algebraic structure](https://bliptext.com/articles/algebraic-structure).\nSemi[rings](https://bliptext.com/articles/ring-algebra) are a generalization of [rings](https://bliptext.com/articles/ring-algebra), dropping the requirement that each element must have an [additive inverse](https://bliptext.com/articles/additive-inverse).\nAt the same time, semirings are a generalization of [bounded](https://bliptext.com/articles/lattice-order) [distributive lattices](https://bliptext.com/articles/distributive-lattice).\nThe smallest semiring that is not a ring is the [two-element Boolean algebra](https://bliptext.com/articles/two-element-boolean-algebra), for instance with [logical disjunction](https://bliptext.com/articles/logical-disjunction) \\lor as addition.\nA motivating example that is neither a ring nor a lattice is the set of [natural numbers](https://bliptext.com/articles/natural-number) \\N (including zero) under ordinary addition and multiplication.\nSemirings are abundant because a suitable multiplication operation arises as the [function composition](https://bliptext.com/articles/function-composition) of [endomorphisms](https://bliptext.com/articles/endomorphism) over any [commutative monoid](https://bliptext.com/articles/commutative-monoid).\n\n# Terminology\nSome authors define semirings without the requirement for there to be a 0 or 1.\nThis makes the analogy between **ring** and on the one hand and and on the other hand work more smoothly.\nThese authors often use **rig** for the concept defined here.\nThis originated as a joke, suggesting that rigs are rings without negative elements.\n(Akin to using [rng](https://bliptext.com/articles/rng-algebra) to mean a ring without a multiplicative identity.)\nThe term **dioid** (for \"double monoid\") has been used to mean semirings or other structures.\nIt was used by Kuntzmann in 1972 to denote a semiring.\n(It is alternatively sometimes used for [naturally ordered semirings](https://bliptext.com/articles/naturally-ordered-semiring) but the term was also used for idempotent subgroups by [Baccelli](https://bliptext.com/articles/fran-ois-baccelli) et al. in 1992.)\n\n# Definition\nA **semiring** is a [set](https://bliptext.com/articles/set-mathematics) R equipped with two [binary operations](https://bliptext.com/articles/binary-operation) + and \\cdot, called addition and multiplication, such that:\nFurther, the following axioms tie to both operations:\n\n## Notation\nThe symbol \\cdot is usually omitted from the notation; that is, a \\cdot b is just written ab.\nSimilarly, an [order of operations](https://bliptext.com/articles/order-of-operations) is conventional, in which \\cdot is applied before +.\nThat is, denotes.\nFor the purpose of disambiguation, one may write 0_R or 1_R to emphasize which structure the units at hand belong to.\nIf x\\in R is an element of a semiring and, then n-times repeated multiplication of x with itself is denoted x^n, and one similarly writes for the n-times repeated addition.\n\n# Construction of new semirings\nThe [zero ring](https://bliptext.com/articles/zero-ring) with underlying set \\{0\\} is a semiring called the trivial semiring.\nThis triviality can be characterized via 0=1 and so when speaking of nontrivial semirings, 0\\neq 1 is often silently assumed as if it were an additional axiom.\nNow given any semiring, there are several ways to define new ones.\nAs noted, the natural numbers {\\mathbb N} with its arithmetic structure form a semiring.\nTaking the zero and the image of the successor operation in a semiring R, i.e., the set together with the inherited operations, is always a sub-semiring of R.\nIf (M, +) is a commutative monoid, function composition provides the multiplication to form a semiring: The set of endomorphisms M \\to M forms a semiring where addition is defined from pointwise addition in M. The [zero morphism](https://bliptext.com/articles/zero-morphism) and the identity are the respective neutral elements.\nIf M = R^n with R a semiring, we obtain a semiring that can be associated with the square n\\times n [matrices](https://bliptext.com/articles/matrix-mathematics) with coefficients in R, the [matrix semiring](https://bliptext.com/articles/matrix-semiring) using ordinary [addition](https://bliptext.com/articles/matrix-addition) and [multiplication](https://bliptext.com/articles/matrix-multiplication) rules of [matrices](https://bliptext.com/articles/matrix-mathematics).\nGiven and R a semiring, is always a semiring also.\nIt is generally non-commutative even if R was commutative.\n[Dorroh extensions](https://bliptext.com/articles/rng-algebra): If R is a semiring, then with pointwise addition and multiplication given by defines another semiring with multiplicative unit.\nVery similarly, if N is any sub-semiring of R, one may also define a semiring on R\\times N, just by replacing the repeated addition in the formula by multiplication.\nIndeed, these constructions even work under looser conditions, as the structure R is not actually required to have a multiplicative unit.\n[Zerosumfree](https://bliptext.com/articles/zerosumfree-monoid) semirings are in a sense furthest away from being rings.\nGiven a semiring, one may adjoin a new zero 0' to the underlying set and thus obtain such a zerosumfree semiring that also lacks [zero divisors](https://bliptext.com/articles/zero-divisor).\nIn particular, now and the old semiring is actually not a sub-semiring.\nOne may then go on and adjoin new elements \"on top\" one at a time, while always respecting the zero.\nThese two strategies also work under looser conditions.\nSometimes the notations -\\infty resp.\n+\\infty are used when performing these constructions.\nAdjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the [logical connectives](https://bliptext.com/articles/logical-connectives) of disjunction and conjunction:.\nConsequently, this is the smallest semiring that is not a ring.\nExplicitly, it violates the ring axioms as for all P, i.e. 1 has no additive inverse.\nIn the [self-dual](https://bliptext.com/articles/duality-order-theory) definition, the fault is with.\n(This is not to be conflated with the ring \\Z_2, whose addition functions as [xor](https://bliptext.com/articles/xor) \\veebar.)\nIn the [von Neumann model of the naturals](https://bliptext.com/articles/set-theoretic-definition-of-natural-numbers),, and.\nThe two-element semiring may be presented in terms of the set theoretic union and intersection as.\nNow this structure in fact still constitutes a semiring when 1_\\omega is replaced by any inhabited set whatsoever.\nThe [ideals](https://bliptext.com/articles/ideal-ring-theory) on a semiring R, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring.\nThe ideals of 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.\nIf R is a semiring and A is an [inhabited set](https://bliptext.com/articles/inhabited-set), A^* denotes the [free monoid](https://bliptext.com/articles/free-monoid) and the formal polynomials R[A^*] over its words form another semiring.\nFor small sets, the generating elements are conventionally used to denote the polynomial semiring.\nFor example, in case of a singleton A=\\{X\\} such that, one writes R[X].\nZerosumfree sub-semirings of R can be used to determine sub-semirings of R[A^*].\nGiven 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.\nGiven a [derivation](https://bliptext.com/articles/derivation-differential-algebra) {\\mathrm d} on a semiring R, another the operation \"\\bullet\" fulfilling can be defined as part of a new multiplication on R[X], resulting in another semiring.\nThe above is by no means an exhaustive list of systematic constructions.\n\n## Derivations\nDerivations on a semiring R are the maps with and.\nFor example, if E is the 2\\times 2 unit matrix and, then the subset of given by the matrices a\\,E + b\\,U with a,b\\in R is a semiring with derivation.\n\n# Properties\nA basic property of semirings is that 1 is not a left or right [zero divisor](https://bliptext.com/articles/zero-divisor), and that 1 but also 0 squares to itself, i.e. these have u^2=u.\nSome notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty.\nAlso, the [2-ary](https://bliptext.com/articles/arity) predicate defined as, here defined for the addition operation, always constitutes the right [canonical](https://bliptext.com/articles/monoid) [preorder](https://bliptext.com/articles/preorder) relation.\n[Reflexivity](https://bliptext.com/articles/reflexive-relation) is witnessed by the identity.\nFurther, is always valid, and so zero is the [least element](https://bliptext.com/articles/least-element) with respect to this preorder.\nConsidering it for the commutative addition in particular, the distinction of \"right\" may be disregarded.\nIn the non-negative integers \\N, for example, this relation is [anti-symmetric](https://bliptext.com/articles/anti-symmetric-relation) and [strongly connected](https://bliptext.com/articles/strongly-connected), and thus in fact a (non-strict) [total order](https://bliptext.com/articles/total-order).\nBelow, more conditional properties are discussed.\n\n## Semifields\nAny **[field](https://bliptext.com/articles/field-mathematics)** is also a semi**[field](https://bliptext.com/articles/field-mathematics)**, which in turn is a semiring in which also multiplicative inverses exist.\n\n## Rings\nAny field is also a **ring**, which in turn is a semi**ring** in which also additive inverses exist.\nNote that a semiring omits such a requirement, i.e., it requires only a [commutative monoid](https://bliptext.com/articles/commutative-monoid), not a [commutative group](https://bliptext.com/articles/commutative-group).\nThe extra requirement for a ring itself already implies the existence of a multiplicative zero.\nThis contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly.\nHere -1, the additive inverse of 1, squares to 1.\nAs additive differences d=y-x always exist in a ring, is a trivial binary relation in a ring.\n\n## Commutative semirings\nA semiring is called a **[commutative](https://bliptext.com/articles/commutative) semiring** if also the multiplication is [commutative](https://bliptext.com/articles/commutative).\nIts axioms can be stated concisely: It consists of two commutative monoids and on one set such that and.\nThe [center](https://bliptext.com/articles/center-ring-theory) of a semiring is a sub-semiring and being commutative is equivalent to being its own [center](https://bliptext.com/articles/center-ring-theory).\nThe commutative semiring of natural numbers is the [initial object](https://bliptext.com/articles/initial-object) among its kind, meaning there is a unique structure preserving map of {\\mathbb N} into any commutative semiring.\nThe bounded distributive lattices are [partially ordered](https://bliptext.com/articles/partial-order), commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence.\nThus so are their [duals](https://bliptext.com/articles/duality-theory-for-distributive-lattices).\n\n## Ordered semirings\nNotions or order can be defined using strict, non-strict or [second-order](https://bliptext.com/articles/second-order-logic) formulations.\nAdditional properties such as commutativity simplify the axioms.\nGiven a [strict total order](https://bliptext.com/articles/strict-total-order) (also sometimes called linear order, or [pseudo-order](https://bliptext.com/articles/pseudo-order) in a constructive formulation), then by definition, the positive and negative elements fulfill 0

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