=====Idempotent semirings with identity=====

Abbreviation: **ISRng$_1$**

====Definition====
An \emph{idempotent semiring with identity} is a [[semirings with identity]] $\mathbf{S}=\langle S,\vee,\cdot,1
\rangle $ such that

$\vee$ is idempotent:  $x\vee x=x$

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

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

====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====

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

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

====Superclasses====
[[Idempotent semirings]] 

[[Semirings with identity]] 


====References====

[(Ln19xx>
)]