Does this equality hold for all lattices with a top?

Question

Let $(L,\land,\lor)$ be a lattice with maximal element $1$.

Let $p_1,...,p_n,q_1,...,q_m \in L: \lor_{i=1}^n p_i=\lor_{j=1}^m q_i =1$.

Is it guaranteed that $\lor_{i=1}^n \lor_{j=1}^m (p_i \land q_j)=1$, or is distributivity required?

Answer 1

It doesn't hold for all lattices.

For example, consider the diamond, $\mathbf M_3$, with middle elements $p_1, q_1$ and $p_2=q_2$.
It follows that $p_1 \vee p_2 = q_1 \vee q_2 = 1$, but $$(p_1\wedge q_1) \vee (p_1 \wedge q_2) \vee (p_2 \wedge q_1) \vee (p_2 \wedge q_2) = 0 \vee 0 \vee 0 \vee p_2 \neq 1.$$

Distributivity is also not strictly necessary, for consider the pentagon, $\mathbf N_5$, with middle elements $p_1 < q_1 || p_2=q_2$ (here, $x || y$ means that $x\nleq y$ and $y \nleq x$).
It follows that $p_1 \vee p_2 = q_1 \vee q_2 = 1$ and $$(p_1\wedge q_1) \vee (p_1 \wedge q_2) \vee (p_2 \wedge q_1) \vee (p_2 \wedge q_2) = p_1 \vee 0 \vee 0 \vee p_2 = 1.$$

Of course the implication holds if the lattice is distributive, but I suppose that is clear to you.