Lattice of congruences of a lattice is distributive

Question

Let $\mathcal{L}=(L;\land,\lor)$ be a lattice. Show that the lattice $(\operatorname{Con}\mathcal{L},\subseteq)$ is distributive.

I understand that I only need to show that for any $a,b \in L$ and congruences $A,B,C$ that $$(a,b ) \in(A\land(B\ \lor C)) \Rightarrow (a,b) \in ((A\land B) \lor (A\land C)).$$

If we take $(a,b) \in (A\land(B\ \lor C))$ then $(a,b) \in A$ and $(a,b) \in (B \lor C)$ and we only need to show that $(a,b) \in B$ or $(a,b) \in C.$

It should be easy, but for some reason I can't prove this. Any help would be greatly appreciated.

Answer 1

HINT: You can’t in general show that $\langle a,b\rangle$ belongs to $B$ or $C$; here’s a counterexample.

Let $L=\Bbb Z$, $m\land n=\gcd(m,n)$, and $m\lor n=\operatorname{lcm}(m,n)$. Let $A$ be congruence mod $2$, $B$ be congruence mod $12$, and $C$ be congruence mod $18$. Then $B\cup C$ is congruence mod $6$, so $\langle 2,8\rangle\in A\land(B\lor C)$, but $\langle 2,8\rangle$ is not in $A\land B$ or in $A\land C$.

For $a,b,c\in L$ define

$$M(a,b,c)=(a\lor b)\land(a\lor c)\land(b\lor c)\;,$$

and note that

$$M(a,a,b)=M(a,b,a)=M(b,a,a)=a$$

for any $a,b\in L$. The existence of such a function $M$ definable within $L$ is sufficient to ensure that $\operatorname{Con}\mathcal{L}$ is distributive.

Suppose that $\langle a,b\rangle\in A\land(B\lor C)$. Use (or prove and then use) the fact that $B\lor C$ is the transitive closure of $B\cup C$ to get $x_0,\ldots,x_{2n}\in L$ such that $x_0=a$, $x_{2n}=b$, and for $k=0,1,\ldots,n-1$ we have $\langle x_{2k},x_{2k+1}\rangle\in B$ and $\langle x_{2k+1},x_{2k+2}\rangle\in C$.

• Verify that $\big\langle M(a,x_k,b),M(a,x_k,a)\big\rangle\in A$ for $k=0,\ldots,2n$. Use the fact that any congruence on $L$ must respect the function $M$.

• Show that $$\big\langle M(a,x_{2k},b),M(a,x_{2k+1},b)\big\rangle\in A\land B$$ and $$\big\langle a,x_{2k+1},b),M(a,x_{2k+2},b)\big\rangle\in A\land C$$ for $k=0,1,\ldots,n-1$.

• Conclude that $\langle a,b\rangle\in(A\land B)\lor(A\land C)$.