Semiring
Algebraic ring that need not have additive negative elements / From Wikipedia, the free encyclopedia
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In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.
The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers under ordinary addition and multiplication, when including the number zero. Semirings are abundant, because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.
The theory of (associative) algebras over commutative rings can be generalized to one over commutative semirings.[citation needed]
Some authors call semiring the structure without the requirement for there to be a or . 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.[1][lower-alpha 1] This originated as a joke, suggesting that rigs are rings without negative elements. (And this is similar 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 Kuntzman in 1972 to denote a semiring.[2] (It is alternatively sometimes used for naturally ordered semirings[3] but the term was also used for idempotent subgroups by Baccelli et al. in 1992.[4])
A semiring is a set equipped with two binary operations and called addition and multiplication, such that:[5][6][7]
- is a monoid with identity element called :
- is a monoid with identity element called :
- Addition is commutative:
Explicitly stated, is a commutative monoid. Further, the following axioms tie to both operations:
- Through multiplication, any element is left- and right-annihilated by the additive identity:
- Multiplication left- and right-distributes over addition:
Notation
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, denotes .
For the purpose of disambiguation, one may write or to emphasize which structure the units at hand belong to.
If is an element of a semiring and , then -times repeated multiplication of with itself is denoted , and one similarly writes for the -times repeated addition.
The zero ring with underlying set is also a semiring, called the trivial semiring. This triviality can be characterized via and so 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 with its arithmetic structure form a semiring. The set equipped with the operations inherited from a semiring , is always a sub-semiring of .
If is a commutative monoid, function composition provides the multiplication to form a semiring: The set of endomorphisms forms a semiring, where addition is defined from pointwise addition in . The zero morphism and the identity are the respective neutral elements. If with a semiring, we obtain a semiring that can be associated with the square matrices with coefficients in , the matrix semiring using ordinary addition and multiplication rules of matrices. Yet more abstractly, given and a semiring, is always a semiring also. It is generally non-commutative even if was commutative.
Dorroh extensions: If is a semiring, then with pointwise addition and multiplication given by :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle } defines another semiring with multiplicative unit . Very similarly, if is any sub-semiring of , one may also define a semiring on , just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure 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 to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular, now 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 resp. 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: . Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as for all , i.e. has no additive inverse. In the self-dual definition, the fault is with . (This is not to be conflated with the ring , whose addition functions as xor .) In the von Neumann model of the naturals, , and . The two-element semiring may be presented in terms of the set theoretic union and intersection as . Now this structure in fact still constitutes a semiring when is replaced by any inhabited set whatsoever.
The ideals on a semiring , with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of are in bijection with the ideals of . The collection of left ideals of (and likewise the right ideals) also have much of that algebraic structure, except that then does not function as a two-sided multiplicative identity.
If is a semiring and is an inhabited set, denotes the free monoid and the formal polynomials 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 such that , one writes . Zerosumfree sub-semirings of can be used to determine sub-semirings of .
Given a set , not necessarily just a singleton, adjoining a default element to the set underlying a semiring one may define the semiring of partial functions from to .
Given a derivation on a semiring , another the operation "" fulfilling can be defined as part of a new multiplication on , resulting in another semiring.
The above is by no means an exhaustive list of systematic constructions.
Derivations
Derivations on a semiring are the maps with and .
For example, if is the unit matrix and , then the subset of given by the matrices with is a semiring with derivation .