> You are welcome to think about the union-find structure however you prefer to think about it, but I was describing the problem I was trying to solve, for which the correct description of union find I gave above is optimal.

> If your way of thinking about union find makes it hard for you to understand the problem I was trying to solve, maybe it isn't the best way to think about it for the purpose of this conversation, even if it is the best way to think about it in some other context.

I'm not having any problems understanding the problem you were trying to solve. My problems are in the area of understanding why you thought a union-find might be an effective way to address that problem.

I'm telling you that, if you adjust your view of what a union-find is, you will have a better conception of where it can and can't be usefully applied.

> In general there is a very close correspondence between binary relations and digraphs, so it's usually easy to reformulate a statement about relations as an equivalent statement about digraphs, and vice versa. But one formulation or the other may be more perspicacious.

In this case, the relation is symmetric by definition, so you just have graphs. Yes, it's obvious how you can use a graph to describe a relation.

But the entire point of the union-find structure is to discard any information contained in an equivalent graph that isn't contained in the relation. (And there are many equivalent graphs, but only one relation!) The absence of that information is what makes the structure valuable. It shouldn't be a surprise that, if you want to preserve information from a particular graph, you are better served with graph-based algorithms.