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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.

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Have you looked in the literature on quantales? A quantale is the same as an idempotent semiring, except that quantales are required to have all (not just finite) joins, and correspondingly multiplication has to distribute over all joins. –  Tom Leinster Mar 20 '12 at 15:57

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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:

  1. arXiv:math/0509684, Toen-Vaquie, Under Spec Z
  2. arXiv:0704.2030, Durov, New Approach to Arakelov Geometry
  3. arXiv:1103.1745, Lorscheid, The geometry of blueprints. Part I: Algebraic background and scheme theory
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