PlanetMath: idempotent semiring

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idempotent semiring

(Definition)

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

$\displaystyle a+a=a,$ for all $\displaystyle a\in S.$

Some properties of an i-semiring .

If we define a binary relation $\le$ on by
$\displaystyle a\le b$ iff $\displaystyle \qquad a+b=b$
then $\le$ becomes a partial order on . Indeed, for implies $a\le a$ ; if $a\le b$ and $b\le a$ , then ; and finally, if $a\le b$ and $b\le c$ , then so $a\le c$ .
$0\le a$ for any $a\in S$ , because .
Define $a\vee b$ as the supremum of and (with respect to $\le$ ). Then $a\vee b$ exists and
$\displaystyle a\vee b=a+b.$
To see this, we have , so $a\le a+b$ . Similarly $b\le a+b$ . If $a\le c$ and $b\le c$ , then . So $a+b\le c$ .
Collecting all the information above, we see that is an upper semilattice with as the join operation on and 0 the bottom element.
Additon and multiplication respect partial ordering: suppose $a\le b$ , then for any $c\in S$ , , hence $c+a\le c+b$ ; also, implies $ca\le cb$ .