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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 xy. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff yx.

I assume that the following holds:

For every hereditarily finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(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}))?
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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 xy. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff yx.

I assume that the following holds:

For every finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(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}))?
4 edited body

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 xy. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff yx.

I assume that the following holds:

For every finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(Y).

Questions

Is this assumption correct? 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. $\lbrace 1, 1'\rbrace$ 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}))?
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