Galois insertion

Remarks

Let $G$ be a group, let $P$ be the set of subsets of $G$, and let $Q$ be the set of subgroups of $G$, both ordered by inclusion. Let $f(S)$ be the subgroup generated by $S$, and let $g(H) = H$ as a subset. Then $f(S) \subseteq H$ if and only if $S \subseteq H$, and $f(g(H)) = H$, so this is a Galois insertion. The pair of the inclusion $\Z \to \R$ and the floor function $\R \to \Z$ is a Galois connection but not a Galois insertion. There the lower adjoint is the inclusion, and the inclusion of $\lfloor 1/2 \rfloor = 0$ is $0 \ne 1/2$. For a Galois connection, these conditions are equivalent: $f \circ g$ is the identity of $Q$; $f$ is surjective; $g$ is injective. The dual notion, with $g(f(x)) = x$ for all $x \in P$, is a Galois coinsertion.