We show that a property known from model theory as weak elimination of imaginaries corresponds to the property of admitting least algebraically closed supports, and is a generalization of admitting least supports in many contexts. In particular, we show that an ω-categorical structure admits least supports if and only if it has no algebraicity and has weak elimination of imaginaries.
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.
This post discusses equivalence of:
1. existence of the least supports
2. representation theorem