# 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 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 .

Proof.

Hypothesis. For every atoms , there is an orbit-finite superset such that the conditions of the above theorem hold with taken as atoms.