Let the *transitive closure graph* of a set *X* be the graph *G(X)* with V(*G*)= TC({*X*}) and (*x,y*) ∈ E(*G*) iff *x* ∈ *y*. Let *H(X)* be the reverse graph of *G(X)* with (*x,y*) ∈ E(*H*) iff *y* ∈ *x*.

I assume that the following holds:

For every hereditarily finite set

Xthere is a unique setYsuch thatG(X)is isomorphic toH(Y).

**Questions**

Is this assumption correct? *[ Edit: If the answer is no: for which X?]* Has this set

*Y*been given a name (as a function of

*X*)? Something like the

*reverse set of X*? And has it attracted some interest? Can someone give a reference?

Some simple facts:

The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (

*self-reverse*for short)The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse.

The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 2'\rbrace$ and $\lbrace 0, 2\rbrace$ vs. $\lbrace 0, 1, 2'\rbrace$.

Further question:

- Is there a general correlation between the cardinalities of
*X*, its reverse set*Y*and TC({*X*}) (= TC({*Y*}))?