Can't you define for any set of partitions of , rather than w.r.t. a specific factorization , simply as iff ? If so, it would seem to me to be clearer to define that way (i.e. make 7 rather than 2 from proposition 10 the definition), and then basically proposition 10 says "if is a subset of factors of a partition then here are a set of equivalent definitions in terms of chimera". Also I would guess that proposition 11 is still true for rather than just for , though I haven't checked that 11.6 would still work, but it seems like it should.
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 F=(S,B), a partition X∈Part(S), and a C⊆B, we say C generates X (in F), written C⊢FX, if χFC(x,S)=x for all x∈X.
The following proposition gives many equivalent definitions of ⊢F.
Proposition 10. Let F=(S,B) be a finite factored set, let X∈Part(S) be a partition of S, and let C be a subset of B. The following are equivalent:
Proof. The equivalence of conditions 1 and 2 is by definition.
The equivalence of conditions 2 and 3 follows directly from the fact that χFC(s,s)=s for all s∈x, so χFC(x,S)⊇χFC(x,x)⊇x.
To see that conditions 3 and 4 are equivalent, observe that since S=⋃y∈Xy, χFC(x,S)=⋃y∈XχFC(x,y). Thus, if χFC(x,S)⊆x, χFC(x,y)⊆x for all y∈X, and conversely if χFC(x,y)⊆x for all y∈X, then χFC(x,S)⊆x.
To see that condition 3 is equivalent to condition 5, observe that if condition 5 holds, then for all x∈X, we have χFC(s,t)∈[s]X=x for all s∈x and t∈S. Thus χFC(x,S)⊆x. Conversely, if condition 3 holds, χFC(s,t)∈χFC([s]X,S)⊆[s]X for all s,t∈S.
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 s,t∈S satisfy s∼⋁S(C)t, then χFC(s,t)=t, so t=χFC(s,t)∼Xs. Thus X≤S⋁S(C). Conversely, if condition 7 holds, then since χFC(s,t)∼⋁S(C)s for all s,t∈S, we have χFC(s,t)∼Xs. □
Here are some basic properties of ⊢F.
Proposition 11. Let F=(S,B) be a finite factored set, let C and D be subsets of B, and let X,Y∈Part(S) be partitions of S.
Proof. For the first 5 parts, we will use the equivalent definition from Proposition 10 that C⊢FX if and only if X≤S⋁S(C).
Then 1 follows directly from the transitivity of ≤S.
2 follows directly from the fact that any partition Z satisfies X∨SY≤Z if and only if X≤Z and Y≤Z.
3 follows directly from the fact that ⋁S(B)=DisS by Proposition 3.
4 follows directly from the fact that ⋁S({})=IndS, together with the fact that X≤SIndS if and only if X=IndS.
5 follows directly from the fact that if C⊆D, then ⋁S(C)≤⋁S(D).
Finally, we need to prove part 6. For this, we will use the equivalent definition from Proposition 10 that C⊢FX if and only if χFC(s,t)∼Xs for all s,t∈S. Assume that for all s,t∈S, χFC(s,t)∼Xs and χFD(s,t)∼Xs. Thus, for all s,t∈S, χFC∩D(s,t)=χFC(χFD(s,t),t)∼XχFD(s,t)∼Xs. Thus C∩D⊢FX. □
Our main use of ⊢F 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 F=(S,B) and a partition X∈Part(S), let hF(X) denote the smallest (according to the subset ordering) subset of B such that hF(X)⊢FX.
The history of X, then, is the smallest set of factors C⊆B such that if you're trying to figure out which part in X any given s∈S is in, it suffices to know what part s is in within each of the factors in C. We can informally think of hF(X) as the smallest amount of information needed to compute X.
Proposition 12. Given a finite factored set F=(S,B), and a partition X∈Part(S), hF(X) is well-defined.
Proof. Fix a finite factored set F=(S,B) and a partition X∈Part(S), and let hF(X) be the intersection of all C⊆B such that C⊢FX. It suffices to show that hF(X)⊢FX; then hF(X) will clearly be the unique smallest (according to the subset ordering) subset of B such that hF(X)⊢FX.
Note that hF(X) is a finite intersection, since there are only finitely many subsets of B, and that hF(X) is an intersection of a nonempty collection of sets since B⊢FX. Thus, we can express hF(X) as a composition of finitely many binary intersections. By part 6 of Proposition 11, the intersection of two subsets that generate X also generates X. Thus hF(X)⊢FX. □
Here are some basic properties of history.
Proposition 13. Let F=(S,B) be a finite factored set, and let X,Y∈Part(S) be partitions of S.
Proof. The first 3 parts are trivial consequences of history's definition and Proposition 11.
For the fourth part, observe that {b}⊢Fb by condition 7 of Proposition 10. b is nontrivial, and since S is nonempty, b is nonempty. So we have ¬({}⊢Fb) by part 4 of Proposition 11. Thus {b} is the smallest subset of B that generates b. □
3.3. Orthogonality
We are now ready to define the notion of orthogonality between two partitions of S.
Definition 18 (orthogonality). Given a finite factored set F=(S,B) and partitions X,Y∈Part(S), we say X is orthogonal to Y (in F), written X⊥FY, if hF(X)∩hF(Y)={}.
If ¬(X⊥FY), we say X is entangled with Y (in F).
We could also unpack this definition to not mention history or chimera functions.
Proposition 14. Given a finite factored set F=(S,B), and partitions X,Y∈Part(S), X⊥FY if and only if there exists a C⊆B such that X≤S⋁S(C) and Y≤S⋁S(B∖C).
Proof. If there exists a C⊆B such that X≤S⋁S(C) and Y≤S⋁S(B∖C), then C⊢FX and B∖C⊢FY. Thus, hF(X)⊆C and hF(Y)⊆B∖C, so hF(X)∩hF(Y)={}.
Conversely, if hF(X)∩hF(Y)={}, let C=hF(X). Then C⊢FX, so X≤S⋁S(C), and B∖C⊇hF(Y), so B∖C⊢FY, so Y≤S⋁S(B∖C). □
Here are some basic properties of orthogonality.
Proposition 15. Let F=(S,B) be a finite factored set, and let X,Y,Z∈Part(S) be partitions of S.
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 F=(S,B), and partitions X,Y∈Part(S), we say X is before Y (in F), written X≤FY, if hF(X)⊆hF(Y).
We say X is strictly before Y (in F), written X<FY, if hF(X)⊂hF(Y).
Again, we could also unpack this definition to not mention history or chimera functions.
Proposition 16. Given a finite factored set F=(S,B), and partitions X,Y∈Part(S), X≤FY if and only if every C⊆B satisfying Y≤S⋁S(C) also satisfies X≤S⋁S(C).
Proof. Note that by part 7 of Proposition 10, part 5 of Proposition 11, and the definition of history, C satisfies Y≤S⋁S(C) if and only if C⊇hF(Y), and similarly for X.
Clearly, if hF(Y)⊇hF(X), every C⊇hF(Y) satisfies C⊇hF(X). Conversely, if hF(X) is not a subset of hF(Y), then we can take C=hF(Y), and observe that C⊇hF(Y) but not C⊇hF(X). □
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 F=(S,B), and partitions X,Y∈Part(S), X≤FY if and only if every Z∈Part(S) satisfying Y⊥FZ also satisfies X⊥FZ.
Proof. Clearly if hF(X)⊆hF(Y), then every Z satisfying hF(Y)∩hF(Z)={} also satisfies hF(X)∩hF(Z)={}.
Conversely, if hF(X) is not a subset of hF(Y), let b∈B be an element of hF(X) that is not in hF(Y). Assuming S is nonempty, b is nonempty, so we have hF(b)={b}, so Y⊥Fb, but not X⊥Fb. On the other hand, if S is empty, then X=Y={}, so clearly X≤FY. □
Here are some basic properties of time.
Proposition 18. Let F=(S,B) be a finite factored set, and let X,Y,Z∈Part(S) be partitions of S.
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 4 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 F=(S,B) and a partition X∈Part(S), hF(X)={b∈B∣b≤FX}.
Proof. Since S is nonempty, part 4 of Proposition 13 says that hF(b)={b} for all b∈B. Thus {b∈B∣b≤FX}={b∈B∣{b}⊆hF(X)}={b∈B∣b∈hF(X)}=hF(X). □
In the next post, we'll build up to a definition of conditional orthogonality by introducing the notion of subpartitions.