Given any function \(S \xrightarrow{g} T\), we can induce a Galois connection \(Prt(S) \leftrightarrows Prt(T)\) between the sets of partitions of the domain and codomain.
Determine the left adjoint \(Prt(S) \xrightarrow{g_!} Prt(T)\)
Starting with a given partition in \(S\), obtain a partition in \(T\) by saying two elements, \(t_1,t_2\) are in the same partition if \(\exists s_1 \sim s_2: g(s_1)=t_1 \land g(s_2)=t_2\)
This is not necessarily a transitive relation, so take the transitive closure.
Determine the right adjoint \(Prt(T) \xrightarrow{g^*} Prt(S)\)
Given a partition of \(T\), we say two elements in \(S\) are connected iff \(g(s_1) \sim g(s_2)\)
Given a function \(\{1 \mapsto 12, 2 \mapsto 12, 3 \mapsto 3, 4 \mapsto 4\}\) from the four element set \(S\) to the three element set \(T\)
Choose a nontrivial partition \(c \in Prt(S)\) and compute \(g_!(c) \in Prt(T)\)
Choose any coarser partition \(g_!(c)\leq d \in Prt(T)\)
Choose any non-coarser partition \(g_!(c) > e \in Prt(T)\)
Find \(g^*(d)\) and \(g^*(e)\)
Show that the adjunction formula is true, i.e. that \(c \leq g^*(d)\) (because \(g_!(c) \leq d\)) and \(g^*(e) > c\) (because \(e > g_!(c)\))
\(c = \{(1, 3),(2,), (4,)\}\), \(g_!(c)\) is then \(\{(12,3),(4,)\}\)
\(d = \{(12,),(3,),(4,)\}\)
\(e = \{(12,3,4)\}\)
\(g^*(d)=\{(1,2),(3,),(4,)\}, g^*(e)=\{(1,2,3,4)\}\)
\(c \leq g^*(d)\) and \(g^*(e) > c\)