I am reading "Characterization of generic extensions of models of set theory" in https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1703/bukovsky.pdf.
Definition 1: For a relation r, let $ r''a \iff $ { $ y \in rng(r) : (\exists x ∈ a) <x, y> \in r $ } [so presumably this is like a Valuation using a]
Definition 2: M[a] is the smallest inner model such that M ⊆ M[a] and a ∈ M[a].
Definition 3: The definition of support arises in Section 3 :
"A set σ ⊆ M is a support over M if for any relations $r_1, r_2 \in M$ there exists a relation r ∈ M such that $r''σ = r''_1 σ$ \ $r''_2 σ .$
If $x = r''σ, r \in M \text{ then x } \in M[σ]$.
If N = M [G], where G is an ultrafilter on a partially ordered set generic over M , then G is a support over M . Actually, for every x ⊆ M , x ∈ M [G], there exists a relation r ∈ M , such that x = r'' G. If G is an ultrafilter on a complete Boolean algebra, then for any such x even $x = f^{−1}(G)$ for some function $f \in M$ ."
So the question is : What is the intuition behind the definition of support ?
[I can see that G is intended to intersect all dense sets D, which can be viewed as sets with a relation:
$r_D$:={ $<x,y> : x \in P \land y \leq x$}.
Also whenever there is $ r''_1 σ$ \ $r''_2 σ $ then a representation of negation comes to mind, so $ P $ \ $G $ could be constructed from some object $ r \in M$, but there must be much more to it ....]