By an idempotent semiring I mean a set equipped with a join-semilattice with bottom structure $(0,+)$ and a multiplicative monoid $(1,\cdot)$ such that the following equations hold:

$a \cdot (b + c) = a \cdot b + a \cdot c$ $\;\;\;\; (a + b) \cdot c = a \cdot c + b \cdot c$ $\;\;\;\; a \cdot 0 = 0 = 0 \cdot a$

Let $\mathcal{A}$ be the category whose objects are the idempotent semirings and whose morphisms are the algebra morphisms i.e. functions preserving the operations. Is there any known concrete characterisation of $\mathcal{A}^{op}$? What about in the case where one restricts to the full subcategory of finitely generated algebras?

Any help much appreciated.