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Concept
Semiring
Summary
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 endomorphism over any commutative monoid.
The theory of (associative) algebras over commutative rings can be generalized to one over commutative semirings.
Some authors call semiring the structure without the requirement for there to be a or . This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. 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. (It is alternatively sometimes used for naturally ordered semirings, but the term was also used for idempotent subgroups by Baccelli et al. in 1992.)
A semiring is a set equipped with two binary operations and called addition and multiplication, such that:
is a monoid with identity element called :
is a monoid with identity element called :
Addition is commutative:
Multiplication by the additive identity annihilates :
Multiplication left- and right-distributes over addition:
Explicitly stated, is a commutative monoid.
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 .
Official source
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