Dual of idempotent semirings 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.
 A: I don't think you can ask for a characterisation of $\mathcal{A}^{op}$ that is any more concrete than the definition. 
However, $\mathcal{A}^{op}$ can be described in alternative terms via scheme theory.  This won't describe it in any simpler terms, and in fact it introduces a good deal of extra complication, but it can perhaps be useful sometimes because it puts things in a more geometric setting. In standard algebraic geometry the category of commutative rings is equivalent to the opposite of the category of affine schemes (over spec $\mathbb{Z}$).   Similarly, your category of idempotent commutative semirings is equivalent to the opposite of the category of affine schemes over spec $S$, where $S$ is the initial object in idempotent semirings ($S=${0,1}).
This embeds as a full subcategory of the category of affine schemes over $\mathbb{N}$ (the semiring of natural numbers).  For references on the scheme theory of semirings, see:


*

*arXiv:math/0509684, Toen-Vaquie, Under Spec Z

*arXiv:0704.2030, Durov, New Approach to Arakelov Geometry

*arXiv:1103.1745, Lorscheid, The geometry of blueprints. Part I: Algebraic background and scheme theory

