Editing Semiring (section)

== Construction of new semirings ==
The [[zero ring]] with underlying set <math>\{0\}</math> is a semiring called the trivial semiring. This triviality can be characterized via <math>0=1</math> and so when speaking of nontrivial semirings, <math>0\neq 1</math> 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 <math>{\mathbb N}</math> with its arithmetic structure form a semiring. Taking the zero and the image of the successor operation in a semiring <math>R</math>, i.e., the set <math>\{x\in R\mid x=0_R\lor \exists p. x = p + 1_R\}</math> together with the inherited operations, is always a sub-semiring of <math>R</math>.

If <math>(M, +)</math> is a commutative monoid, function composition provides the multiplication to form a semiring: The set <math>\operatorname{End}(M)</math> of endomorphisms <math>M \to M</math> forms a semiring where addition is defined from pointwise addition in <math>M</math>. The [[zero morphism]] and the identity are the respective neutral elements. If <math>M = R^n</math> with <math>R</math> a semiring, we obtain a semiring that can be associated with the square <math>n\times n</math> [[Matrix (mathematics)|matrices]] <math>{\mathcal M}_n(R)</math> with coefficients in <math>R</math>, the [[matrix semiring]] using ordinary [[Matrix addition|addition]] and [[Matrix multiplication|multiplication]] rules of matrices. Given <math>n\in{\mathbb N}</math> and <math>R</math> a semiring, <math>{\mathcal M}_n(R)</math> is always a semiring also. It is generally non-commutative even if <math>R</math> was commutative.

[[Rng (algebra)#Adjoining an identity element (Dorroh extension)|Dorroh extensions]]: If <math>R</math> is a semiring, then <math>R\times{\mathbb N}</math> with pointwise addition and multiplication given by <math>\langle x,n\rangle\bullet \langle y,m\rangle:=\langle x\cdot y+(x\,m+y\,n), n\cdot m\rangle</math> defines another semiring with multiplicative unit <math>1_{R\times{\mathbb N}}:=\langle 0_R,1_{\mathbb N}\rangle</math>. Very similarly, if <math>N</math> is any sub-semiring of <math>R</math>, one may also define a semiring on <math>R\times N</math>, just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure <math>R</math> is not actually required to have a multiplicative unit.

[[Zerosumfree monoid|Zerosumfree]] semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero <math>0'</math> to the underlying set and thus obtain such a zerosumfree semiring that also lacks [[zero divisor]]s. In particular, now <math>0\cdot 0'=0'</math> 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 <math>-\infty</math> resp. <math>+\infty</math> 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: <math>\langle\{0,1\},+,\cdot,\langle 0,1\rangle\rangle=\langle\{\bot,\top\},\lor,\land,\langle\bot,\top\rangle\rangle</math>. Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as <math>\top\lor P = \top</math> for all <math>P</math>, i.e. <math>1</math> has no additive inverse. In the [[Duality (order theory)|self-dual]] definition, the fault is with <math>\bot\land P = \bot</math>. (This is not to be conflated with the ring <math>\Z_2</math>, whose addition functions as [[xor]] <math>\veebar</math>.) 
In the [[Set-theoretic definition of natural numbers|von Neumann model of the naturals]], <math>0_\omega:=\{\}</math>, <math>1_\omega:=\{0_\omega\}</math> and <math>2_\omega:=\{0_\omega,1_\omega\}={\mathcal P}1_\omega</math>. The two-element semiring may be presented in terms of the set theoretic union and intersection as <math>\langle {\mathcal P}1_\omega,\cup,\cap,\langle \{\},1_\omega\rangle\rangle</math>. Now this structure in fact still constitutes a semiring when <math>1_\omega</math> is replaced by any inhabited set whatsoever.

The [[Ideal (ring theory)|ideals]] on a semiring <math>R</math>, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of <math>{\mathcal M}_n(R)</math> are in bijection with the ideals of <math>R</math>. The collection of left ideals of <math>R</math> (and likewise the right ideals) also have much of that algebraic structure, except that then <math>R</math> does not function as a two-sided multiplicative identity.

If <math>R</math> is a semiring and <math>A</math> is an [[inhabited set]], <math>A^*</math> denotes the [[free monoid]] and the formal polynomials <math>R[A^*]</math> 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 <math>A=\{X\}</math> such that <math>A^*=\{\varepsilon,X,X^2,X^3,\dots\}</math>, one writes <math>R[X]</math>. Zerosumfree sub-semirings of <math>R</math> can be used to determine sub-semirings of <math>R[A^*]</math>.

Given a set <math>A</math>, not necessarily just a singleton, adjoining a default element to the set underlying a semiring <math>R</math> one may define the semiring of partial functions from <math>A</math> to <math>R</math>.

Given a [[Derivation (differential algebra)|derivation]] <math>{\mathrm d}</math> on a semiring <math>R</math>, another the operation "<math>\bullet</math>" fulfilling <math>X\bullet y=y\bullet X+{\mathrm d}(y)</math> can be defined as part of a new multiplication on <math>R[X]</math>, resulting in another semiring.

The above is by no means an exhaustive list of systematic constructions.

=== Derivations ===
Derivations on a semiring <math>R</math> are the maps <math>{\mathrm d}\colon R\to R</math> with <math>{\mathrm d}(x+y)={\mathrm d}(x)+{\mathrm d}(y)</math> and <math>{\mathrm d}(x\cdot y)={\mathrm d}(x)\cdot y+x\cdot {\mathrm d}(y)</math>.

For example, if <math>E</math> is the <math>2\times 2</math> unit matrix and <math>U=\bigl(\begin{smallmatrix}0 & 1 \\ 0 & 0 \end{smallmatrix}\bigr)</math>, then the subset of <math>{\mathcal M}_2(R)</math> given by the matrices <math>a\,E + b\,U</math> with <math>a,b\in R</math> is a semiring with derivation <math>a\,E + b\,U\mapsto b\,U</math>.