<p>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.</p>

CC BY-SA 3.0
2016-03-11T16:53:53.747Z
2016-03-11T16:53:53.747Z
Emil Jeřábek
<p>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 $x\le_2 y\le_2 x$).</p>

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

<p>Can I conclude anything about the relationship between my preorders?</p>

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

<partial-order><order-theory>
Two preorders with same glb
Uli Fahrenberg
# Two preorders with same glb

(Asked by: username_0 on 2016-03-11)

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 $\mathord{\le}\_1=\mathord{\le}\_2$.)

**Tags:** partial-order, order-theory

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## 1 answers

### Answer by username_1 on 2016-03-11. Score: 3 | Accepted answer

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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