What are some examples of commutative semirings such that the following hold?
- Multiplication is idempotent i.e. we have $xx=x$ for all elements $x$.
- Addition is not idempotent i.e. there is at least one element $x$ with $x+x \neq x$.
- There is at least one element $x$ with $x+x \neq 0$.
I cannot think of any examples, with or without commutativity.
Discussion. In ring theory, it is a theorem that if multiplication is idempotent, then $x+x=0$ for all elements $x$. However, in semiring theory, this isn't necessarily the case; take, in particular, any non-trivial distributive lattice. Then it is certainly the case that $x+x \neq 0$ for all $x,$ except $x=0$ of course. In fact, we have that $x+x=x$ for all elements $x$.
The initial semiring is $\mathbb{N}$ (with the usual operations). Hence, the initial boolean semiring is $R = \mathbb{N} / \sim$, where $\sim$ is the smallest congruence relation such that $n \sim n^2$ for all $n \in \mathbb{N}$. Hence, we also have $n_1^2 + \dotsc + n_s^2 \sim n_1 + \dotsc + n_s$. The elements of $R$ are therefore $$[0],[1],[2],[3],[4]=[2],[5]=[3],[6]=[2],[7]=[3],\dotsc$$ We find that $R$ is actually finite, it has exactly $4$ elements. This ring satisfies your requirements.
Another example is the free boolean semiring on one generator $x$. It is given by $\{a+bx : a,b \in R\}$ modulo the relation $a+bx \sim (a+bx)^2 = (a+b)+(2ab)x$. This semiring is also finite and its elements may be listed explicitly, if one wants to.