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.
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Least supports and the representation theorem
This post discusses equivalence of:
1. existence of the least supports
2. representation theorem
Fact. Suppose that is a one-orbit set and
is an equivariant function. Then is a bijection.