Semirings with zero

Abbreviation: SRng$_0$

Definition

A semiring with zero is a structure $\mathbf{S}=\langle S,+,0,\cdot \rangle $ of type $\langle 2,0,2\rangle $ such that

$\langle S,+,0\rangle $ is a commutative monoid

$\langle S,\cdot\rangle$ is a semigroup

$0$ is a zero for $\cdot$: $0\cdot x=0$, $x\cdot 0=0$

$\cdot$ distributes over $+$: $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$

Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with zero. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$

Subclasses

Superclasses

References