# Every Distributive Lattice is an Inserter of Boolean Algebras

Proposition 10 of (Dahlqvist and Kurz, 2017) shows that every (bounded) distributive lattice is an inserter of Boolean algebras.

Here is an example how the 3-chain arises as the set
$$\{x\in B \mid in_1(x)\le in_2(x)\}$$

where $B$ is the 4-element BA and $in_1$ and $in_2$ are BA-morphisms from $B$ to the 8-element BA $\mathcal P(\{x,y,z\})$ so that $in_1(a)=\{x\}\subseteq\{x,y\}=in_2(a)$.


It is also interesting to draw the dual diagram in the category of posets. In fact, this is how I first found the upper picture. Indeed, it is easy to see that every poset is a coinserter of discrete posets: One just needs to use that the order is a binary relation on the set in question.


Note that $\{x,y,z\}$ together with $in_1^\partial, in_2^\partial$ is (isomorphic to) the order relation $a\le 1$ if we identify $x=(a,a)$, $y=(a,1)$ and $z=(1,1)$. This gives us $a=in_1^\partial(y)\le in_2^\partial(y)=1$.

## References

Dahlqvist and Kurz: [The Positivication of Coalgebraic Logics](https://drops.dagstuhl.de/opus/volltexte/2017/8042/pdf/LIPIcs-CALCO-2017-9.pdf). 2017.