=====Semirings with identity and zero=====

Abbreviation: **SRng**$_{01}$

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


$\langle S,+,0\rangle $ is a [[commutative monoids]]


$\langle S,\cdot,1\rangle$ is a [[monoids]]


$0$ 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 identity and zero. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |no |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &   1\\
f(2)= &   2\\
f(3)= &   6\\
f(4)= &  40\\
f(5)= & 295\\
f(6)= &3246\\
\end{array}$

====Subclasses====
[[Idempotent semirings with identity and zero]] 

[[Rings with identity]] 

====Superclasses====
[[Semirings with zero]] 

[[Semirings with identity]] 


====References====

[(Ln19xx>
)]