The linked paper introduces the key concept of factored spaced models / finite factored sets, structural independence, in a fully general setting using families of random elements. The key contribution is a general definition of the history object and a theorem that the history fully characterizes the semantic implications of the assumption that a family of random elements is independent. This is analogous to how d-separation precisely characterizes which nodal variables are independent given some nodal variables in any probability distribution which fulfills the markov property on the graph.

Abstract: Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to d-separation and structural causal models.

Formally, let $U=(U_i{)}_{i\in I}$ be an independent family of random elements on a probability space $(\Omega ,A,P)$. Let $X$, $Y$, and $Z$ be arbitrary $\sigma \left(U\right)$-measurable random elements. We characterize all independences $X\perp \perp Y\mid Z$ implied by the independence of $U$ and call these independences structural. Formally, these are the independences which hold in all probability measures $P$ that render $U$ independent and are absolutely continuous with respect to $P$, i.e., for all such $P$, it must hold that $X{\perp }_{P}Y\mid Z$.

We introduce the history $\mathrm{history}(X\mid Z):\Omega \to P\left(I\right)$, a combinatorial object that measures the dependence of $X$ on $U_i$ for each $i\in I$ given $Z$. The independence of $X$ and $Y$ given $Z$ is implied by the independence of $U$ if and only if $\mathrm{history}(X\mid Z)\cap \mathrm{history}(Y\mid Z)=\varnothing$ almost surely with respect to $P$.

Finally, we apply this d-separation-like criterion in structural causal models to discover a causal direction in a toy setting.