It can happen that two equivariant sets with atoms are related by a finitely supported bijection, but not by an equivariant bijection.
All posts by Bartek
Orbit finiteness vs. number of supported elements
A set is orbit finite if and only if the number of its elements supported by 
atoms grows polynomially with .
The Rum Conjecture is refuted
A conjecture I posted a while ago, concerning reachability of states in automata with atoms after a Brzozowski phase, has been refuted by Mikołaj Bojańczyk.
Existence of least supports for Fraisse atoms
In Automata theory in nominal sets (Thm. 9.3), we provided a necessary and sufficient criterion for the existence of least finite supports for arbitrary atoms. It was not clear whether the criterion could be significantly simplified for Fraisse atoms. I show that it cannot be made as simple as we originally conjectured.
A conjecture concerning Brzozowski algorithm (PRIZE!)
Fix attention on equality atoms.
Given an orbit-finite, deterministic automaton on the alphabet of atoms, a Brzozowski phase amounts to:
- reversing the direction of all transition arrows and swapping initial and accepting states,
- determinising the result, and
- cutting to the reachable part.
A pumping lemma for automata with atoms
This is a straightforward generalization of the classical Pumping Lemma for regular languages. It was proved independently by the Warsaw team and by Filippo Bonchi, Daniela Petrisan and Alexandra Silva.