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I'm currently reading Dugundji's Topology's book. Exercise 4 in chapter 2 says (omitting a part that is irrelevant to the question)

A partially order set is a lattice if each pair of its elements has a least upper bound and a greatest lower bound. Is $(P(X),\subseteq)$ a lattice? In $P(A\times A),\subseteq)$, is the set of all transitive relations a lattice? Determine also if the set of all partial orders, preorders, and well orders are each lattices.

This is what I got. $P(X)$ is clearly a lattice cause the supremum of $A$ and $B$ is clearly $A\cup B$ and the infimum is $A\cap B$. The set of all transitive relations is a lattice cause given two relations $R$, $S$, the relation $T=\bigcap\{Q:R\cup S\subseteq Q \text{ and Q is a transitive relation}\}$ is a transitive relation that is also the supremum of $A$ and $B$ (note also that this intersection is non-empty cause $A\times A$ is transitive). The same trick works for preorders. I'm struggling with partial orders and well orders, the problem is that $A\times A$ is not in general a partial order (let alone a well order) so I cannot prove that the intersection is nonempty. For the infimum one can always take the intersection $R\cap S$.

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  • $\begingroup$ I just realized that, edited. $\endgroup$ – Zero Jan 18 '15 at 0:45
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Hint for partial orders and well orders: Consider concretely $A=\{1,2\}$ and the orders $<$ and $>$. Do they have a common upper bound at all (let alone a least upper bound)?

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