idempotent semiring

A semiring $S$ is called an idempotent semiring, or i-semiring for short, if, addition $+$ is an idempotent binary operation:

a+a=a,\qquad\mbox{ for all }a\in S.

Some properties of an i-semiring $S$ .

1.

If we define a binary relation $\leq$ on $S$ by

$a\leq b\qquad\mbox{ iff }\qquad a+b=b$

then $\leq$ becomes a partial order on $S$ . Indeed, for $a+a=a$ implies $a\leq a$ ; if $a\leq b$ and $b\leq a$ , then $b=a+b=a$ ; and finally, if $a\leq b$ and $b\leq c$ , then $a+c=a+(b+c)=(a+b)+c=b+c=c$ so $a\leq c$ .
2.

$0\leq a$ for any $a\in S$ , because $0+a=a$ .
3.

Define $a\vee b$ as the supremum of $a$ and $b$ (with respect to $\leq$ ). 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 $a\leq a+b$ . Similarly $b\leq a+b$ . If $a\leq c$ and $b\leq c$ , then $(a+b)+c=a+(b+c)=a+c=c$ . So $a+b\leq c$ .
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 $a\leq b$ , then for any $c\in S$ , $(c+a)+(c+b)=(c+c)+(a+b)=c+b$ , hence $c+a\leq c+b$ ; also, $cb=c(a+b)=ca+cb$ implies $ca\leq cb$ .

Remark. $S$ in general is not a lattice, and $1$ is not the top element of $S$ .

The main example of an i-semiring is a Kleene algebra used in the theory of computations.