I have a set $S$ with two preorders $\mathord{\le}_1,\mathord{\le}_2\subseteq S\times S$ which a priori are unrelated. Let $\equiv_1$ and $\equiv_2$ be the induced equivalences (i.e., $x\equiv_1 y$ iff $x\le_1 y\le_1 x$ and $x\equiv_2 y$ iff $x\le_2 y\le_2 x$).

I also have an operation $\land$ which turns out to give greatest lower bounds for both preorders: for all $x,y\in S$, $x\le_1 y$ iff $x\land y\equiv_1 x$ and $x\le_2 y$ iff $x\land y\equiv_2 x$.

Can I conclude anything about the relationship between my preorders?

(Note that the problem becomes trivial if my preorders are partial orders: in that case, the above implies $\mathord{\le}_1=\mathord{\le}_2$.)


I'd say no. Consider the following example. Let $(A,\land)$ and $(B,\land)$ be two unrelated semilattices. Let $(S,\land)$ be their direct product, and put $(a,b)\le_1(a',b')$ iff $a\le a'$, $(a,b)\le_2(a',b')$ iff $b\le b'$. Then your conditions are satisfied, even though the two preorders look nothing like each other.

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