Given a binary relation $R \subseteq X \times Y$, we get an antitone Galois connection $(F,U) : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ in the usual way:
The function $U : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ is given by $U(A) = \{y ∈ Y \,|\, \forall a \in A, (a,y) \in R\}.$
The function $F : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)$ is given by $F(B) = \{x ∈ X \,|\, \forall b \in B, (b,x) \in R^\dagger\}.$
($R^\dagger$ denotes the converse of $R$.)
I'm interested in a special case of this. Assume that $X$ equals $Y,$ and that $R$ is a symmetric relation. That is, $R=R^\dagger$. Then $U$ equals $F,$ so lets write $C$ for either/both of $U,F$. In this case, let us say that $C$ is the induced connection of $R.$
We can say it a little differently. By graph, let us mean a set (not necessarily finite) equipped with a symmetric binary relation (not necessarily reflexive or irreflexive). Then to every graph $(X,R)$, there is an induced connection $C : \mathcal{P}(X) \rightarrow \mathcal{P}(X)$.
Question. Where can I learn more about the Galois connection induced by a graph on its own powerset?
Here's a couple of everyday examples.
Example 0. To every every inner product space $H$, we can assign a relation $R \subseteq H \times H$ defined by asserting that $(x,y) \in R$ iff $\langle x,y\rangle = 0.$ So the previous discussion applies. The induced connection of $R$ is the function $\mathcal{P}(H) \rightarrow \mathcal{P}(H)$ that maps every $A \subseteq H$ to its orthogonal complement $A^\perp \subseteq H$. This particular connection has the interesting property that $A \cap A^\perp = \{0\}$.
Example 1. To every group $G,$ there is a relation $R \subseteq G \times G$ defined by asserting that $(x,y) \in R$ iff $x$ commutes with $y$. Then $R$ is symmetric. The induced connection of $R$ is the function $C : \mathcal{P}(G) \rightarrow \mathcal{P}(G)$ that maps every $A \subseteq G$ to its centralizer $C(A) \subseteq X$. Note in this case that $A \cap C(A)$ needn't equal $\{1\}$, in contrast to the previous example. Anyway, with a bit of thought, this example can be generalized so that $X$ is allowed to be an arbitrary abstract clone, or equivalently a Lawvere theory. This is the case of main interest for me.
You may have a look at the paper "A Primer on Galois Connections" by Erné et al. (see the preprint here), where an antitone Galois connection from $\mathcal{P}(X)$ to $\mathcal{P}(Y)$ is called a polarity. You will find therein many examples and references, including the link with formal concept analysis. However it does not deal specifically with the case where $R$ is symmetric.