Then the relation ✁bo is a *strict total order* on Eo. Intuitively, the equivalence classes in Eo can always be totally ordered using ✁bo, where for any two equivalence classes A and B in Eo, if A ✁bo B, then all event occurrences in A are observed before all the event occurrences in B within the observation o.

For examples, the equivalence classes of the stratified order from Fig. 2.3 can be totally ordered by the ordering ✁bo as follows:
{o1,o2} ✁bo {o3} ✁bo {o4,o5,o6} ✁bo {o7,o8} ✁bo {o9,o10}
When the cardinality of the set of event occurrences is finite as in our example, the stratified order from Fig. 2.3 can be equivalently represented more compactly as

## {O1,O2}{O3}{O4,O5,O6}{O7,O8}{O9,O10},

where each equivalence class is called a *step* and the whole sequence is called a *step* sequence.

It might seem counterintuitive that our axioms allow observations whose infinitely many event occurrences are observed simultaneously. However, this is just a limitation of first order theory. Since our theory allows models that observe arbitrarily large finite set of simultaneous event occurrences, by the compactness theorem there will be models whose observations will allow us to observe infinite set of simultaneous event occurrences.

Observation soundness We have just discussed the idea behind why stratified orders are used to formalize the notion of an observation. We next want to show the intuition of how stratified order based observations satisfy the observation soundness properties with respect to a gso-structure. We will do so using a detailed example.

Example 2. Given the set of event occurrences {oi| 1 ≤ i ≤ 7} and the relations earlier than, not later than and nonsimultaneous from Example 1, we want to know possible observations of this gso-structure. By axioms (2.21) and (2.22) for observation soundness, we know that all of the observations must satisfy all the causality constraints specified by these three relations. For each observation ob, we let Gob denote the dag representing the stratified order ✁ob.

1. The observation ob satisfies the not later than relation intuitively meaning that Gob must contain G1, i.e., G1 ⊆ Gob.

2. The observation ob satisfies the not later than relation *roughly* which means that Gob might or might not contains the edges of G2 − G1, where G2 − G1 denotes the graph difference of G2 and G1. The exception is when G2 − G1 contains both directed edges (oi,oj) and (oj,oi), then neither (oi,oj) nor (oj,oi) is allowed to be included in Gob.

3. Finally ob satisfies the nonsimultaneous relation is equivalent to saying that if
{oi,oj} ∈ G3, but neither (oi,oj) nor (oj,oi) is in the graph G1, then we have the case that either (oi,oj) or (oj,oi) must be included in Gob.