I've been trying to consider a relation construction from pre-existing relations similiar to the construction of product space from pre-existing topological space.
Let $\{ R_\alpha \}_{\alpha\in \Lambda}$ be a collection of homogeneous relation $R_\alpha$ on a set $X_\alpha$, i.e. $R_\alpha \subseteq X_\alpha\times X_\alpha$. We can construct a homogeneous product relation on $\prod_{\alpha \in \Lambda} X_\alpha$ by $R':=\prod_{\alpha\in \Lambda} R_\alpha$. I think that an analogue to the product topology would be a union of 'cylindrical relations' as follows:
For a finite subset $F\subset \Lambda$, I define a homogeneous relation on $ \prod_{\alpha \in \Lambda} X_\alpha $ by
$R'_F:= \prod_{\alpha\in F} R_\alpha \times \prod_{\beta \notin F} \big( X_\beta \times X_\beta \big)$, I then define my new product relation by
$$ \oplus_{\alpha \in \Lambda} R_\alpha := \cup_{F\subset \Lambda, \vert F\vert<\infty} R'_F .$$
My question is whether this construction is standard, and has people encountered this relation somewhere? And if so is there any literature discussing properties of this construction?