This post discusses equivalence of:
1. existence of the least supports
2. representation theorem
Fix atoms, a homogeneous structure over a finite relational vocabulary. A subset of atoms is algebraically closed if all finite -orbits of atoms are included in . Note that algebraically closed sets are always finite.
Theorem. The following conditions are equivalent:
1. Least supports: Every element of a set has the least algebraically closed support. (In other words, finite algebraically closed supports of an element are closed under intersection.)
2. Representation: Every one-orbit set is isomorphic (i.e. related by an equivariant bijection) to a set of the form , where is an orbit of the set of non-repeating tuples of atoms, and is a group of permutations of .
3. Canonical supports: For every one-orbit set there is an equivariant function
with the property that supports , for every .
Hypothesis. For every atoms , there is an orbit-finite superset such that the conditions of the above theorem hold with taken as atoms.