idempotent semiring

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

$$a+a=a,\text{ for all }a\in S.$$

Some properties of an i-semiring $S$.

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 i-semiring is a Kleene algebra used in the theory of computations.