Semirings with identity

Abbreviation: SRng$_1$

Definition

A semiring with identity is a structure $\mathbf{S}=\langle S,+,\cdot,1 \rangle $ of type $\langle 2,2,0\rangle $ such that

$\langle S,+\rangle $ is a commutative semigroup

$\langle S,\cdot, 1\rangle$ is a monoid

$\cdot$ 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:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &11\\ f(4)= &73\\ f(5)= &703\\ f(6)= &\\ \end{array}$

Subclasses

Superclasses

References