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This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...

Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$. One may also consider ...