Semiring

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

The term rig is also used occasionally—this originated as a joke, suggesting that rigs are rings without negative elements, similar to using rng to mean a ring without a multiplicative identity.

A semiring is a set R equipped with two binary operations + and ⋅, called addition and multiplication, such that:

This last axiom is omitted from the definition of a ring: it follows from the other ring axioms. Here it does not, and it is necessary to state it in the definition.

The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily a commutative group. Specifically, elements in semirings do not necessarily have an inverse for the addition.

The symbol ⋅ is usually omitted from the notation; that is, a⋅b is just written ab. Similarly, an order of operations is accepted, according to which ⋅ is applied before +; that is, a + bc is a + (bc).

A commutative semiring is one whose multiplication is commutative. An idempotent semiring is one whose addition is idempotent: a + a = a, that is, (R, +, 0) is a join-semilattice with zero.

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.

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