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.

Atompress

Least supports and the representation theorem

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Computation with atoms