Strong Fraïssé Limits

Martin Ziegler

Freiburg, 30. September 2011

Let K be a countable class of L-structures and S a countable class of embeddings between elements of K which is closed under composition. We call the elements of S strong embeddings. We assume that the empty structure 0 is in K and all 0 → A are strong. (This is a pure convention).

Definition 1. 1. We call a sequence A_{0} → A_{1} → A_{2} → … rich, if the A_{i} are in K, the maps are strong and the following holds: For all i and for all strong f : A_{i} → B there is a j ≥ i and a strong g : B → A_{j} such that gf is the given map A_{i} → A_{j}.

2. A Fraïssé limit of (K, S) is a direkt limit of a rich sequence.^{1}

Theorem 2. If K has the amalgamation property with respect to strong embeddings, rich sequences exist and the Fraïssé limits are all isomorphic.

Proof. Assume amalgamation. The existence of rich sequences is obvious. Now let A_{0} → A_{1} → … and A′_{0} → A′_{1} → … be rich with direkt limits M and M′. We call a partial isomorphisms bewetween M and M′ good if it is given by a pair of strong embeddings e : C → A_{i} and e′ : C → A′_{i′}. We try to extend such an partial isomorphism so that its image contains A_{i}^{2}. This will show the forth property of the family of good isomorphism. By symmetry we will have also the back property.

We amalgamate first e and e′ and obtain strong embeddings f : A_{i} → B and f′ : A′_{i′} → B, such that fe = f′e′. Then we apply richness and get j ≥ i and j′ ≥ i′ and strong embeddings h : B → A_{j} and h′ : B → A′_{j′} such that hf is the given map A_{i} → A_{j} and h′f′ is the given map A′_{i′} → A′_{j′}. This is our prolongation.

The assumption that § is closed under composition is not necessary by the following simple remark.

Remark 3. Let S be a countable set of embeddings between elements of K such that (K, S) has the amalgamation property. Let S′ be the closure of S under composition. Then S′ is countable and (K, S′) still has the amalgamation property

1Note that the direkt limit of a sequence is only defined up to isomorphism.

2or rather the image of A_{i} in M. Note also that we can alway prolong e to a strong map from C to A_{k} for arbitrary k > i