Finite Factored Sets: Orthogonality and Time

The main way we’ll be using factored sets is as a foundation for talking about concepts like orthogonality and time. Finite factored sets will play a role that’s analogous to that of directed acyclic graphs in Pearlian causal inference.

To utilize factored sets in this way, we will first want to introduce the concept of generating a partition with factors.

3.1. Generating a Partition with Factors

Definition 16 (generating a partition). Given a finite factored set , a partition , and a , we say generates (in ), written , if for all .

The following proposition gives many equivalent definitions of .

Proposition 10. Let be a finite factored set, let be a partition of , and let be a subset of . The following are equivalent:

  1. .

  2. for all .

  3. for all .

  4. for all .

  5. for all .

  6. for all .

  7. .

Proof. The equivalence of conditions 1 and 2 is by definition.

The equivalence of conditions 2 and 3 follows directly from the fact that for all , so .

To see that conditions 3 and 4 are equivalent, observe that since , . Thus, if , for all , and conversely if for all , then .

To see that condition 3 is equivalent to condition 5, observe that if condition 5 holds, then for all , we have for all and . Thus . Conversely, if condition 3 holds, for all .

Condition 6 is clearly a trivial restatement of condition 5.

To see that conditions 6 and 7 are equivalent, observe that if condition 6 holds, and satisfy , then , so . Thus . Conversely, if condition 7 holds, then since for all , we have .

Here are some basic properties of .

Proposition 11. Let ) be a finite factored set, let and be subsets of , and let be partitions of .

  1. If and , then .

  2. If and , then .

  3. .

  4. if and only if .

  5. If and , then .

  6. If and , then .

Proof. For the first 5 parts, we will use the equivalent definition from Proposition 10 that if and only if .

Then 1 follows directly from the transitivity of .

2 follows directly from the fact that any partition satisfies if and only if and .

3 follows directly from the fact that by Proposition 3.

4 follows directly from the fact that , together with the fact that if and only if .

5 follows directly from the fact that if , then .

Finally, we need to prove part 6. For this, we will use the equivalent definition from Proposition 10 that if and only if for all . Assume that for all , and . Thus, for all , . Thus .

Our main use of will be in the definition of the history of a partition.

3.2. History

Definition 17 (history of a partition). Given a finite factored set and a partition , let denote the smallest (according to the subset ordering) subset of such that .

The history of , then, is the smallest set of factors such that if you’re trying to figure out which part in any given is in, it suffices to know what part is in within each of the factors in . We can informally think of as the smallest amount of information needed to compute .

Proposition 12. Given a finite factored set , and a partition , is well-defined.

Proof. Fix a finite factored set and a partition , and let be the intersection of all such that . It suffices to show that ; then will clearly be the unique smallest (according to the subset ordering) subset of such that .

Note that is a finite intersection, since there are only finitely many subsets of , and that is an intersection of a nonempty collection of sets since . Thus, we can express as a composition of finitely many binary intersections. By part 6 of Proposition 11, the intersection of two subsets that generate also generates . Thus .

Here are some basic properties of history.

Proposition 13. Let be a finite factored set, and let be partitions of .

  1. If , then .

  2. .

  3. if and only if .

  4. If is nonempty, then for all .

Proof. The first 3 parts are trivial consequences of history’s definition and Proposition 11.

For the fourth part, observe that by condition 7 of Proposition 10. is nontrivial, and since is nonempty, is nonempty. So we have by part 4 of Proposition 11. Thus is the smallest subset of that generates .

3.3. Orthogonality

We are now ready to define the notion of orthogonality between two partitions of .

Definition 18 (orthogonality). Given a finite factored set and partitions , we say is orthogonal to (in ), written , if .

If , we say is entangled with (in ).

We could also unpack this definition to not mention history or chimera functions.

Proposition 14. Given a finite factored set , and partitions , if and only if there exists a such that and .

Proof. If there exists a such that and , then and . Thus, and , so .

Conversely, if , let . Then , so , and , so , so .

Here are some basic properties of orthogonality.

Proposition 15. Let be a finite factored set, and let be partitions of .

  1. If , then .

  2. If and , then .

  3. If and , then .

  4. if and only if .

Proof. Part 1 is trivial from the symmetry in the definition.

Parts 2, 3, and 4 follow directly from Proposition 13.

3.4. Time

Finally, we can define our notion of time in a factored set.

Definition 19 ((strictly) before). Given a finite factored set , and partitions , we say is before (in ), written , if .

We say is strictly before (in ), written , if .

Again, we could also unpack this definition to not mention history or chimera functions.

Proposition 16. Given a finite factored set , and partitions , if and only if every satisfying also satisfies .

Proof. Note that by part 7 of Proposition 10, part 5 of Proposition 11, and the definition of history, satisfies if and only if , and similarly for .

Clearly, if , every satisfies . Conversely, if is not a subset of , then we can take , and observe that but not .

Interestingly, we can also define time entirely as a closure property of orthogonality. We hold that the philosophical interpretation of time as a closure property on orthogonality is natural and transcends the ontology set up in this sequence.

Proposition 17. Given a finite factored set , and partitions , if and only if every satisfying also satisfies .

Proof. Clearly if , then every satisfying also satisfies .

Conversely, if is not a subset of , let be an element of that is not in . Assuming is nonempty, is nonempty, so we have , so , but not . On the other hand, if is empty, then , so clearly .

Here are some basic properties of time.

Proposition 18. Let be a finite factored set, and let be partitions of .

  1. .

  2. If and , then .

  3. If , then .

  4. If and , then .

Proof. Part 1 is trivial from the definition.

Part 2 is trivial by transitivity of the subset relation.

Part 3 follows directly from part 1 of Proposition 13.

Part 3 follows directly from part 2 of Proposition 13.

Finally, note that we can (circularly) redefine history in terms of time, thus partially justifying the names.

Proposition 19. Given a nonempty finite factored set and a partition , .

Proof. Since is nonempty, part 4 of Proposition 13 says that for all . Thus .

In the next post, we’ll build up to a definition of conditional orthogonality by introducing the notion of subpartitions.