In his recent post Bartek demonstrated an example of two equivariant sets with atoms that are related by a finitely supported bijection, but not by any equivariant bijection. We show that if the sets are orbit-finite then this situation is impossible.
Proposition. Let and be two orbit-finite equivariant sets with atoms. If there exists a finitely supported bijection then there exists an equivariant bijection .
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Let be two elements with atoms. The pair satisfies the following universal property: supports both and , and if is another element with this property, then supports .
We demonstrate the existence of an element denoted , with a dual property: is supported both by and by , and if is another element with this property, then is supported by . The construction does not require any assumptions on the symmetry.
It can happen that two equivariant sets with atoms are related by a finitely supported bijection, but not by an equivariant bijection.
In the equality symmetry, there is a single-orbit graph with the property that is bipartite in the classical sense (equivalently, has no odd-length cycle) but has no partition into two parts which are finitely supported and not inter-connected. In other words, the graph maps to the single-edge graph via a infinitely-supported homomorphism, but not via a finitely-supported one. And here’s the graph: its nodes are pairs of distinct atoms, and there is an edge from to , whenever are distinct atoms. That’s it.
Let be an equivariant relation. We show that under a certain assumption on , for every , if there exists a witness such that holds, then there also exists one with small support. We show an application to extending homomorphisms.