Least supports and the representation theorem

This post discusses equivalence of:

1. existence of the least supports

2. representation theorem

Fix atoms,  a homogeneous structure \mathbb A over a finite relational vocabulary. A subset S \subseteq {\mathbb A} of atoms is algebraically closed if all finite S-orbits of atoms are included in S. 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 O / G, where O \subseteq {\mathbb A}^{(n)} is an orbit of the set of non-repeating tuples of atoms, and G is a group of permutations of \{1 \ldots, n\}.

3. Canonical supports:  For every one-orbit set A there is an equivariant function

    \[ s : A \to {\cal P}_\text{fin}(\mathbb A) \]

with the property that f(a) supports a, for every a \in A.

Proof.

 

Hypothesis. For every atoms \mathbb A, there is an orbit-finite superset {\mathbb B} \supseteq {\mathbb A} such that the conditions of the above theorem hold with {\mathbb B} taken as atoms.

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