# Congruences on the pentagon lattice $\mathcal{N}_5$

Let $$\mathcal{N}_5$$ refer to the Pentagon lattice, or the lattice generated by the set $$\{0, a, b, c, 1\}$$ subject to $$1 > a$$, $$1 > c$$, $$a > b$$, $$b > 0$$ and $$c > 0$$.

My aim is to find the congruences on $$\mathcal{N}_5$$. I know that in any lattice, a congruence class under a congruence relation is a convex sublattice. Thus, one approach I could take is enumerating all convex sublattices of $$\mathcal{N}_5$$ and trying to see which of those correspond to congruence classes under various congruences on the lattice.

However, this seems like a lot of enumeration, and I feel like there must be a more elegant way to approach this problem (consider, by contrast, the ease of characterizing congruences on $$n$$-element chains $$C_n$$). So do I truly just have to bite the bullet and start enumerating, or is there something else I could do?

I think it's easier to just enumerate the congruences by hand.

• The only congruence with $$1\sim 0$$ is the trivial congruence.

• If $$b\sim c$$ then $$a\sim b\sim c\sim 1$$ and so we only have one nontrivial congruence identifying $$b$$ and $$c$$.

• We can collapse $$a$$ and $$b$$ without collapsing anything else; the result is then the "diamond" lattice, whose congruence lattice you can compute separately.

• There are two congruences satisfying $$a\sim c$$ and $$b\not\sim c$$, depending on whether or not $$b$$ and $$0$$ are identified.

Through this sort of analysis (really the above is most of the argument), you can determine the entire congruence lattice of $$\mathcal{N}_5$$ without too much trouble.

If you want to be more methodical, you can also think about building the congruence lattice "layer by layer" - e.g. for the first layer, identify for each pair of distinct elements $$(x,y)$$ what is the smallest congruence containing that pair. There aren't too many of these to work through.

• I think you might have consider a different labelling than the OP, since $b\sim c$ or $a\sim c$ implies $0\sim 1$ and everything is related. The lattice of congruences of $N_5$ can be found, for example in this question. Commented Jun 13 at 10:51