Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse.
The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction
Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.
[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.
is a commutative monoid, function composition provides the multiplication to form a semiring: The set
The two-element semiring may be presented in terms of the set theoretic union and intersection as
, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring.
For small sets, the generating elements are conventionally used to denote the polynomial semiring.
, not necessarily just a singleton, adjoining a default element to the set underlying a semiring
, for example, this relation is anti-symmetric and strongly connected, and thus in fact a (non-strict) total order.
The commutative semiring of natural numbers is the initial object among its kind, meaning there is a unique structure preserving map of
Likewise, given any (non-strict) total order, its negation is irreflexive and transitive, and those two properties found together are sometimes called strict quasi-order.
The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily coincide with "
A semiring such that there is a lattice structure on its underlying set is lattice-ordered if the sum coincides with the meet,
The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units).
Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures.
With its standard addition and multiplication, this structure forms the strictly ordered field that is Dedekind-complete.
In particular, the positive additive difference existence can be expressed as Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them.
The non-negative elements of a commutative, discretely ordered ring always validate the axioms of
So a slightly more exotic model of the theory is given by the positive elements in the polynomial ring
plus mathematical induction gives a theory equivalent to first-order Peano arithmetic
However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory.
, which removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom
[13] For commutative, additively idempotent and simple semirings, this property is related to residuated lattices.
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).
tropical semirings on the reals are often used in performance evaluation on discrete event systems.
These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.
However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead.
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.