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Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very reminiscent of the relation between ideals in a polynomial ring and affine algebraic varieties (i won't go more specific in the definitions).

It goes as follows (everything quoted from the above paper): Let $P$ be a finite poset (can be generalized to locally finite), and consider its incidence algebra $I(P,K)$, consisting of all the functions from the intervals in $P$ to some field $K$ (of characteristic zero). Sum and product by scalars are inherited from $K$, and product of two functions $f,g \in P$ is defined as the convolution: \begin{equation} (f*g)(x,y)=\sum_{x\leq z \leq y}f(x,z)g(z,y) \end{equation}

Some special elements in $I(P,K)$ needed to state the connection are the units:

\begin{equation} \delta_{x,y}(u,v)=\begin{cases}1&\text{if $u=x$ and $v=y$,}\\\\0&\text{otherwise.}\end{cases} \end{equation}

Now, the (two sided) ideals in this algebra and the varieties have a very nice relation just very similar to the one from commutative algebra and algebraic geometry. But in this case the relation is tighter, because varieties have an algebraic structure coming from a natural partial ordering.

Define the support of and ideal $J$, $\Delta (J)$, as the set of all the units $\delta_{x,y}$ belonging to $J$. It turns out that every ideal $J$ in $I(P,K)$ consists of all the functions $f$ for which $f(x,y)=0$ whenever $\delta_{x,y}\notin \Delta(J) $. On the other hand, define $Z(J)$ as the set of all intervals $\[ x,y \]$ such that $f(x,y)=0$ for all $f\in J$ (this would be the variety). $Z(J)$ is an order ideal of the poset of all intervals of $P$ (ordered by inclusion).

Theorem: Let $P$ be a finite poset and $S(P)$ the poset of its intervals, ordered by inclusion. Then there is a natural anti-isomorphism between the lattice of ideals of $I(P,K)$ and the lattice of order ideals of $S(P)$.

(For more details and background, check the paper, or "Enumerative Combinatorics Vol.1" by Stanley)

My question is: Does anyone know if this ideal-variety duality has been exploited or studied further in the context of posets from an algebraic geometry point of view? (apart from the material in the mentioned paper).

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The question is very interesting but I am not sure it is well reflected by the title... –  Qiaochu Yuan Aug 25 '11 at 22:43
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Joyal and Tierney have done Algebraic Geometry with lattices (in topoi) somewhat along the lines you describe in their landmark monograph An extension of the Galois Theory of Grothendieck

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That's a really hard book to find... –  Gonçalo Marques Aug 25 '11 at 23:39
    
That's unfortunately true. You can find accounts of the topos theoretic results both in Johnstone's "Sketches of an elephant" and in Borceux/Janelidze's "Galois Theories", but none of these two mentions the "Algebraic Geometry with lattices" point of view. –  Peter Arndt Aug 26 '11 at 9:37
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