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# Duality of Finite Posets and Distributive Lattices

Every distributive lattice be represented as the lattice of upsets of a poset.

Conversely, the logic of upsets of a poset is a (complete) distributive lattice.

To construct the points of a distributive lattice $A$, one can take maximal consistent positive theories, that is, prime filters, or $\sf DL$-morphisms $A\to 2$. They inherit the pointwise order from 2. If we consider $A\to 2$ as the characteristic function of a prime filter, then the pointwise order coincides with the inclusion-order.

In this picture, we have a duality on two levels.
- On the level of homomorphisms, a monotone function $X\to Y$ acts contravariantly, via precomposition, on upsets $2^Y\to 2^X$.
- Representing a distributive lattice $A$ as a lattice of upsets of a poset $X$ via a $\sf DL$-morphism $\sem{-}:A\to 2^X$ amounts to specifying a relation
$${\Vdash}: X\times A \to 2$$

If we take $\sem{-}$ to be an inclusion, $\Vdash$ is the elementship relation. The fact that $\Vdash$ is monotone in $X$ corresponds to the elements of $A$ being up-closed subsets of $X$.

Alternatively, $\sem{-}:A\to 2^{X^o}$ represents $A$ as down-closed subsets of $X$, corresponding to a "weakening relation" ${\Vdash}: X^o\times A \to 2$. Then $X\to[A,2]^o$ represents $X$ as prime filters of $A$ ordered by reverse inclusion.

## Example

Consider the lattice $D$ generated by $m \le a$ and $m\le b$ and $1\le a\vee b$.


1. The set of join-irreducible elements of $D$ is $X=\{m,a,b\}$. Moreover, $X$ is a poset with the order inherited from $D$.


We can recover $D$ from $X$ as the set $\mathcal DX=[X^o,2]$ of downsets of $X$.

Writing $\sem{-}$ for the isomorphism $D\to\mathcal DX$, the logical interpretation of this duality is
$$x \models d \ \Leftrightarrow x\in \sem d.$$

The set of join-irreducible elements is order-isomorphic to the set of prime filters ordered by reverse-inclusion. We can also work with prime filteres ordered by inclusion:

2. Let $X$ now be the set of prime filteres ordered by inclusion. This $X$ is order-dual to the one of item 1. Then $D$ is isomorphic to the set $[X,2]$ of upsets of $X$ ordered by inclusion.

The logical interpretation is
$$x \models d \ \Leftrightarrow d\in x.$$

In other words, the "semantic world" $x$ is identified with its theory $\{d \mid x\models d\}$. In our example, these worlds are ordered as follows.


We can think of this order as an information order. In the world $x_m$ both $a$ and $b$ hold, whereas in $x_a$ only $a$ but not $b$ holds. This order is also known as the *specialisation order* defined as
$$x\le x' \ \stackrel {\rm def} \Longleftrightarrow \ \forall d\,. x\models d \Rightarrow x'\models d.$$

It coincides with theory inclusion, or also, with inclusion of prime filters.

Another possibility to set up the duality is by working with meet-irreducibles. In distributive lattices this gives us another isomorphic account of this duality:

- The set of meet-irreducible elements of $D$ is $Y=\{a,b,0\}$. Again, $Y$ is a poset with the order inherited from $D$. Then $D$ is isomorphic to the set $\mathcal UX$ of upsets of $X$, where $\mathcal UX$ is ordered by reverse-inclusion.

The set of meet-irreducible elements is order-isomorphic to the of prime ideals ordered by inclusion.

## Theory

The duality between finite posets and finite distributive lattices is known as Birkhoff's representation theorem, see eg Theorems 5.9 and 5.12 of (Davey and Priestley, 2002).

Davey and Priestley formulate this duality, as we have done in item 1, in terms of join-irreducibles and downsets. The equivalent formulation of item 2 arises from the observation that $\mathcal DX$ is order-isomorphic to $(\mathcal UX)^{op}$. [^DXUX] Thus, instead of defining


in terms of join-irreducibles and downsets, we can also use prime filters and upsets, both ordered by inclusion. The disadvantage of this is that it amounts to equipping the poset $X$ of the introductory example with its dual order. The advantage is that on the level of functors we have $P=[-,2]$ and $S=[-,2]$. Moreover, we have seen that the order on $X$ then has a natural interpretation as information order.

## References

Davey and Priestley: Introduction to Lattices and Order (2nd ed.), 2002. (Thm 5.9 and 5.12)

[^DXUX]: $\mathcal DX=[X^{o},2]$ is the set of downsets ordered by inclusion, $\mathcal UX = [X,2]^{o}$ is the set of upsets ordered by reverse inclusion. The isomorphism $\mathcal DX\cong(\mathcal UX)^{o}$ is mediated by the order-isomorphism $swap:2\to 2^{o}$ as indicated in the following picture.


This makes use of the fact that upsets ordered by reverse inclusion correspond to arrows $X^{o}\to 2^{o}$.