idempotent semiring
A semiring $S$ is called an idempotent semiring, or isemiring for short, if, addition^{} $+$ is an idempotent^{} binary operation^{}:
$$a+a=a,\text{for all}a\in S.$$ 
Some properties of an isemiring $S$.

1.
If we define a binary relation^{} $\le $ on $S$ by
$$a\le b\mathit{\hspace{1em}\hspace{1em}}\text{iff}\mathit{\hspace{1em}\hspace{1em}}a+b=b$$ then $\le $ becomes a partial order^{} on $S$. Indeed, for $a+a=a$ implies $a\le a$; if $a\le b$ and $b\le a$, then $b=a+b=a$; and finally, if $a\le b$ and $b\le c$, then $a+c=a+(b+c)=(a+b)+c=b+c=c$ so $a\le c$.

2.
$0\le a$ for any $a\in S$, because $0+a=a$.

3.
Define $a\vee b$ as the supremum^{} of $a$ and $b$ (with respect to $\le $). Then $a\vee b$ exists and
$$a\vee b=a+b.$$ To see this, we have $a+(a+b)=(a+a)+b=a+b$, so $a\le a+b$. Similarly $b\le a+b$. If $a\le c$ and $b\le c$, then $(a+b)+c=a+(b+c)=a+c=c$. So $a+b\le c$.

4.
Collecting all the information above, we see that $(S,+)$ is an upper semilattice^{} with $+$ as the join operation^{} on $S$ and $0$ the bottom element.

5.
Additon and multiplication respect partial ordering: suppose $a\le b$, then for any $c\in S$, $(c+a)+(c+b)=(c+c)+(a+b)=c+b$, hence $c+a\le c+b$; also, $cb=c(a+b)=ca+cb$ implies $ca\le cb$.
Remark. $S$ in general is not a lattice^{}, and $1$ is not the top element of $S$.
The main example of an isemiring is a Kleene algebra used in the theory of computations.
Title  idempotent semiring 

Canonical name  IdempotentSemiring 
Date of creation  20130322 15:52:12 
Last modified on  20130322 15:52:12 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16Y60 
Synonym  isemiring 
Synonym  dioid 