Just as adjunctions give rise to closure operators, from every closure operator we may construct an adjunction.
Let \(P \xrightarrow{j} P\) be a closure operator.
Get a new preorder by looking at a subset of \(P\) fixed by \(j\).
\(fix_j\) defined as \(\{p \in P\ |\ j(p)\cong p\}\)
Define a left adjoint \(P \xrightarrow{j} fix_j\) and right adjoint \(fix_j \xhookrightarrow{g} P\) as simply the inclusion function.
To see that \(j \dashv g\), we need to verify \(j(p) \leq q \iff p \leq q\)
Show \(\rightarrow\):
Because \(j\) is a closure operator, \(p \leq j(p)\)
\(j(p) \leq q\) implies \(p \leq q\) by transivity.
Show \(\leftarrow\):
By monotonicity of \(j\) we have \(p \leq q\) implying \(j(p) \leq j(q)\)
\(q\) is a fix point, so the RHS is congruent to \(q\), giving \(j(p) \leq q\).