Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. Let $\theta$ a congruence of $(S, \lor_{S\times S}, \land_{S\times S})$. Is it true that there is a congruence $\lambda$ of $(L, \lor , \land )$ s.t. $\theta = \lambda \cap S^2$?
I have been unable to tackle this question appropriately. My intuition tells me there should be such congruence and that I must find its construction. I considered letting $\lambda = \theta \cup \{(x, x) : x \notin S\}$; i.e. letting $\lambda$ be the equivalence where all non-members of $S$ are equivalent only to themselves.
To prove that any $\lambda$ we construct is a congruence we should prove that, given $x_0, x_1 \notin S, y_0, y_1 \in S$,
\begin{equation*} (x_0 \circ y_0) \lambda (x_1 \circ y_1) \end{equation*}
where $\circ \in \{\land, \lor \}$. But the $\lambda$ I have suggested tells us nothing about the supremum/infimum between members and non-members of $S$, so evidently it is a mistake.
How should one approach this problem?
