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Noah Schweber
Noah Schweber
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My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as living in a maximal chain of type $\omega$, $\omega^*$, $\zeta$, or $n$ for some finite (even) $n$ and then reason by cases. This argument proves more than just the theorem itself: it shows that a bijection between $A$ and $B$ can be found within $f\cup g^{-1}$.

Now say that a structure $\mathfrak{A}$ (in the sense of universal algebra) is zigzag-intermediate iff

  • for every isomorphic disjoint $\mathfrak{B}$ and every pair of embeddings $f:\mathfrak{A}\rightarrow\mathfrak{B}, g:\mathfrak{B}\rightarrow\mathfrak{A}$ there is an isomorphism $h:\mathfrak{A}\cong\mathfrak{B}$ contained (set-theoretically) in the transitive symmetric closure of $f\cup g^{-1}$ as a relation on $\mathfrak{A}\sqcup\mathfrak{B}$, but

  • it is not the case that such an $h$ can always be found as a subset of $f\cup g^{-1}$.

My question is: do zigzag-intermediate structures exist?

This answer of Pace Nielsen suggests that structures which have any amount of "zigzag-ness" are very close to being sets (to my current reading, anyways); this question is an attempt to test this. My current instinct is that the answer should be no, and a positive answer would change how I interpret Pace's answer.

My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as living in a maximal chain of type $\omega$, $\omega^*$, $\zeta$, or $n$ for some finite (even) $n$ and then reason by cases. This argument proves more than just the theorem itself: it shows that a bijection between $A$ and $B$ can be found within $f\cup g^{-1}$.

Now say that a structure $\mathfrak{A}$ (in the sense of universal algebra) is zigzag-intermediate iff

  • for every isomorphic disjoint $\mathfrak{B}$ and every pair of embeddings $f:\mathfrak{A}\rightarrow\mathfrak{B}, g:\mathfrak{B}\rightarrow\mathfrak{A}$ there is an isomorphism $h:\mathfrak{A}\cong\mathfrak{B}$ contained (set-theoretically) in the transitive symmetric closure of $f\cup g^{-1}$ as a relation on $\mathfrak{A}\sqcup\mathfrak{B}$, but

  • it is not the case that such an $h$ can always be found as a subset of $f\cup g^{-1}$.

This answer of Pace Nielsen suggests that structures which have any amount of "zigzag-ness" are very close to being sets (to my current reading, anyways); this question is an attempt to test this.

My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as living in a maximal chain of type $\omega$, $\omega^*$, $\zeta$, or $n$ for some finite (even) $n$ and then reason by cases. This argument proves more than just the theorem itself: it shows that a bijection between $A$ and $B$ can be found within $f\cup g^{-1}$.

Now say that a structure $\mathfrak{A}$ (in the sense of universal algebra) is zigzag-intermediate iff

  • for every isomorphic disjoint $\mathfrak{B}$ and every pair of embeddings $f:\mathfrak{A}\rightarrow\mathfrak{B}, g:\mathfrak{B}\rightarrow\mathfrak{A}$ there is an isomorphism $h:\mathfrak{A}\cong\mathfrak{B}$ contained (set-theoretically) in the transitive symmetric closure of $f\cup g^{-1}$ as a relation on $\mathfrak{A}\sqcup\mathfrak{B}$, but

  • it is not the case that such an $h$ can always be found as a subset of $f\cup g^{-1}$.

My question is: do zigzag-intermediate structures exist?

This answer of Pace Nielsen suggests that structures which have any amount of "zigzag-ness" are very close to being sets (to my current reading, anyways); this question is an attempt to test this. My current instinct is that the answer should be no, and a positive answer would change how I interpret Pace's answer.

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Noah Schweber
Noah Schweber
  • 21.1k
  • 10
  • 110
  • 331

Cantor-Bernstein phenomena for structures (and a "moderate zigzag" property)

My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as living in a maximal chain of type $\omega$, $\omega^*$, $\zeta$, or $n$ for some finite (even) $n$ and then reason by cases. This argument proves more than just the theorem itself: it shows that a bijection between $A$ and $B$ can be found within $f\cup g^{-1}$.

Now say that a structure $\mathfrak{A}$ (in the sense of universal algebra) is zigzag-intermediate iff

  • for every isomorphic disjoint $\mathfrak{B}$ and every pair of embeddings $f:\mathfrak{A}\rightarrow\mathfrak{B}, g:\mathfrak{B}\rightarrow\mathfrak{A}$ there is an isomorphism $h:\mathfrak{A}\cong\mathfrak{B}$ contained (set-theoretically) in the transitive symmetric closure of $f\cup g^{-1}$ as a relation on $\mathfrak{A}\sqcup\mathfrak{B}$, but

  • it is not the case that such an $h$ can always be found as a subset of $f\cup g^{-1}$.

This answer of Pace Nielsen suggests that structures which have any amount of "zigzag-ness" are very close to being sets (to my current reading, anyways); this question is an attempt to test this.