Then we have nonsimultaneousM =
T✁∈Ω ✁⇆ and not later thanM =
T✁∈Ω ✁⌢.

⊓⊔
From this theorem, we know that there is always a subset of Ω, where we can uniquely reconstruct not later thanM and nonsimultaneousM . Note that although the consequence of the theorem does not mention earlier thanM , the axiom (2.11) implies that earlier thanM = not later thanM ∩nonsimultaneousM .

Thus, earlier thanM =
T✁∈Ω ✁⇆∩
T✁∈Ω ✁⌢=
T✁∈Ω ✁. Hence, observation completeness is a safe assumption for our gso-structure theory.

It is worth noticing that, since Theorem 1 is a generalization of Szpilrajn's Theorem, the proof of Theorem 1 requires the axiom of choice.

Example 3. Let ✁a, ✁b, ✁c and ✁d be the stratified orders whose transitive reduction graphs are depicted in cases (a), (b), (c) and (d) respectively. Then the set of all the stratified order extensions of the gso-structure from Example 1 is Ω = {✁a,✁b,✁c,✁d}.

However, the gso-structure from Example 1 can be uniquely reconstructed from any subset of Ω, which is a superset of at least one of the following two sets {✁a,✁d} and {✁b,✁c}.

For example, let us consider the set {✁a,✁d}. Then the relations ✁⌢
aand ✁⌢
dcan be represented as the following two graphs (some arcs which can be inferred from transitivity are omitted for simplicity):

![0_image_0.png](0_image_0.png)

It is easy to check that the graph G2 is exactly the intersection of these two graphs. It is also easy to check that the graph G3 is the intersection of the comparability graphs induced by the relations ✁a and ✁d. 

Let Tgso denote our gso-structure theory, which consists of axioms from (2.1) to
(2.24). Then we have the following theorem.

Theorem 2. The theory Tgso is consistent.

$${\mathcal{M}}=\bigcap_{\mathbb{Q}\in\mathbb{Z}}$$
