consider two hypergraphs $H_1 = (V, \mathscr{E}_1), H_2 = (V, \mathscr{E}_2)$ over the same vertex set $V$. am interested in what could be called a "cartesian join" operation building a new hypergraph $H_3=H_1 \sqcup H_2 = (V, \mathscr{E}_1 \times \mathscr{E}_2)$ where "$\times$" is the cartesian join over edge sets. am using the "$\sqcup$" join operator basically as defined in [1], a web page by B.Lynn about ZDDs (a variant of BDD, binary decision diagrams), but havent seen that notation used in a paper so far. BDDs/ZDDs are an equivalent representation system for set families or hypergraphs. ZDDs generally more efficiently model *"sparse"* BDDs without all variable references.

this cartesian join appears not to have been studied much or be much related to hypergraph products (of which there is significant recent material & research) because hypergraph products are all defined over two different vertex sets $V_1, V_2$ and are built out of the cartesian join of the *vertex sets* (*not* the edge sets).

Knuth has enthusiastically investigated ZDDs [2],[3] & has dedicated a significant section of his *Art of computer programming vol4.1* to the subject (30pp and 70 exercises):

*Such operations form a "family algebra," and there are interesting algorithms to implement the operations of family algebra as operations on ZDDs.
Family algebra is a relatively new topic that is just beginning to be understood.*

this cartesian join operation in ZDDs appears to have been 1st implemented by Minato, an originator of ZDD operations. see p10 of [4] where it is called the "cartesian product set P*Q" and was apparently 1st introduced in [6] p7.

what are some other references on or applications of this hypergraph cartesian join operation?

am particularly interested in its similarities to products and factoring incl decomposition algorithms etc.

[1] Families of sets/multiple families by B.Lynn on ZDDs

[2] Katayanagi prize, ZDD Structures and Families of Sets, Donald E. Knuth

[3] Fun with ZDDs talk by Knuth

[4] ZDD and its applications to intelligent processing slides by Minato

[5] Recent and future work on decision diagrams and discrete structure manipulation by Minato

[6] Zero suppressed BDDs and their applications by Minato