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.