# Finite Factored Sets: Orthogonality and Time

The main way we'll be using [factored sets](https://www.lesswrong.com/s/kxs3eeEti9ouwWFzr) 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
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**Definition 16** (generating a partition). *Given a finite factored set *$F=(S,B)$*, a partition *$X\in \mathrm{Part}(S)$*, and a *$C\subseteq B$*, we say *$C$* generates *$X$* (in *$F$*), written *$C\vdash^F X$*, if *$\chi^F_C(x,S)=x$* for all *$x\in X$*.*

The following proposition gives many equivalent definitions of $\vdash^F$.

**Proposition 10**. *Let *$F=(S,B)$* be a finite factored set, let *$X\in \mathrm{Part}(S)$* be a partition of *$S$*, and let *$C$* be a subset of *$B$*. The following are equivalent:*

1.  $C\vdash^F X$*.*
2.  $\chi^F_C(x,S)=x$* for all *$x\in X$*.*
3.  $\chi^F_C(x,S)\subseteq x$* for all *$x\in X$*.*
4.  $\chi^F_C(x,y)\subseteq x$* for all *$x,y\in X$*.*
5.  $\chi^F_C(s,t) \in [s]_X$* for all *$s,t\in S$*.*
6.  $\chi^F_C(s,t)\sim_X s$* for all *$s,t\in S$*.*
7.  $X\leq_S\bigvee_S(C)$*.*

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

The equivalence of conditions 2 and 3 follows directly from the fact that $\chi^F_C(s,s)=s$ for all $s\in x$, so $\chi^F_C(x,S)\supseteq \chi^F_C(x,x)\supseteq x$.

To see that conditions 3 and 4 are equivalent, observe that since $S=\bigcup_{y\in X} y$, $\chi^F_C(x,S)=\bigcup_{y\in X}\chi^F_C(x,y)$. Thus, if $\chi^F_C(x,S)\subseteq x$, $\chi^F_C(x,y)\subseteq x$ for all $y\in X$, and conversely if $\chi^F_C(x,y)\subseteq x$ for all $y\in X$, then $\chi^F_C(x,S)\subseteq x$.

To see that condition 3 is equivalent to condition 5, observe that if condition 5 holds, then for all $x\in X$, we have $\chi^F_C(s,t)\in [s]_X=x$ for all $s\in x$ and  $t\in S$. Thus $\chi^F_C(x,S)\subseteq x$. Conversely, if condition 3 holds, $\chi^F_C(s,t)\in \chi^F_C([s]_X,S)\subseteq [s]_X$ for all $s,t\in S$.

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 $s,t\in S$ satisfy $s\sim_{\bigvee_S(C)}t$, then $\chi^F_C(s,t)=t$, so $t=\chi^F_C(s,t)\sim_X s$. Thus $X\leq_S\bigvee_S(C)$. Conversely, if condition 7 holds, then since $\chi^F_C(s,t)\sim_{\bigvee_S(C)} s$ for all $s,t\in S$, we have $\chi^F_C(s,t)\sim_{X} s$. $\square$

Here are some basic properties of $\vdash^F$.

**Proposition 11**. *Let *$F=(S,B$*) be a finite factored set, let *$C$* and *$D$* be subsets of *$B$*, and let *$X,Y\in \mathrm{Part}(S)$* be partitions of *$S$*.*

1.  *If *$X\leq_S Y$* and *$C\vdash^F Y$*, then *$C\vdash^F X$*.*
2.  *If *$C\vdash^F X$* and *$C\vdash^F Y$*, then *$C\vdash^F X\vee_S Y$*.*
3.  $B\vdash^F X$*.*
4.  $\{\}\vdash^F X$* if and only if *$X=\text{Ind}_S$*.*
5.  *If *$C\subseteq D$* and *$C\vdash^F X$*, then *$D\vdash^F X$*.*
6.  *If *$C\vdash^F X$* and *$D\vdash^F X$*, then *$C\cap D\vdash^F X$*.*

*Proof.* For the first 5 parts, we will use the equivalent definition from Proposition 10 that $C\vdash^F X$ if and only if $X\leq_S\bigvee_S(C)$.

Then 1 follows directly from the transitivity of $\leq_S$.

2 follows directly from the fact that any partition $Z$ satisfies $X\vee_S Y\leq Z$ if and only if $X\leq Z$ and $Y\leq Z$.

3 follows directly from the fact that $\bigvee_S(B)=\text{Dis}_S$ by [Proposition 3](https://www.lesswrong.com/posts/sZa5LQg6rrWgMR4Jx/finite-factored-sets-introduction-and-factorizations#2_2__Factorizations).

4 follows directly from the fact that $\bigvee_S(\{\})=\text{Ind}_S$, together with the fact that $X\leq_S \text{Ind}_S$ if and only if $X=\text{Ind}_S$.

5 follows directly from the fact that if $C\subseteq D$, then $\bigvee_S(C)\leq \bigvee_S(D)$.

Finally, we need to prove part 6. For this, we will use the equivalent definition from Proposition 10 that $C\vdash^F X$ if and only if $\chi^F_C(s,t)\sim_X s$ for all $s,t\in S$. Assume that for all $s,t\in S$, $\chi^F_C(s,t)\sim_X s$ and $\chi^F_D(s,t)\sim_X s$. Thus, for all $s,t\in S$, $\chi^F_{C\cap D}(s,t)=\chi^F_C(\chi^F_D(s,t),t)\sim_X\chi^F_D(s,t)\sim_X s$. Thus $C\cap D\vdash^F X$. $\square$

Our main use of $\vdash^F$ will be in the definition of the history of a partition.

3.2. History
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**Definition 17** (history of a partition). *Given a finite factored set *$F=(S,B)$* and a partition *$X\in \mathrm{Part}(S)$*, let *$h^F(X)$* denote the smallest (according to the subset ordering) subset of *$B$* such that *$h^F(X)\vdash^F X$*.*

The history of $X$, then, is the smallest set of factors $C \subseteq B$ such that if you're trying to figure out which part in $X$ any given $s \in S$ is in, it suffices to know what part $s$ is in within each of the factors in $C$. We can informally think of $h^F(X)$ as the smallest amount of information needed to compute $X$.

**Proposition 12**. *Given a finite factored set *$F=(S,B)$*, and a partition *$X\in\mathrm{Part}(S)$*, *$h^F(X)$* is well-defined.*

*Proof.* Fix a finite factored set $F=(S,B)$ and a partition $X\in\mathrm{Part}(S)$, and let $h^F(X)$ be the intersection of all $C\subseteq B$ such that $C\vdash^F X$. It suffices to show that $h^F(X)\vdash^F X$; then $h^F(X)$ will clearly be the unique smallest (according to the subset ordering) subset of $B$ such that $h^F(X)\vdash^F X$.

Note that $h^F(X)$ is a finite intersection, since there are only finitely many subsets of $B$, and that $h^F(X)$ is an intersection of a nonempty collection of sets since $B\vdash^F X$. Thus, we can express $h^F(X)$ 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 $h^F(X)\vdash^F X$. $\square$

Here are some basic properties of history.

**Proposition 13**. *Let *$F=(S,B)$* be a finite factored set, and let *$X,Y\in \mathrm{Part}(S)$* be partitions of *$S$*.*

1.  *If *$X\leq_S Y$*, then *$h^F(X)\subseteq h^F(Y)$*.*
2.  $h^F(X\vee_S Y)=h^F(X)\cup h^F(Y)$*.*
3.  $h^F(X)=\{\}$* if and only if *$X=\text{Ind}_S$*.*
4.  *If *$S$* is nonempty, then *$h^F(b)=\{b\}$* for all *$b\in B$*.*

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

For the fourth part, observe that $\{b\}\vdash^F b$ by condition 7 of Proposition 10. $b$ is nontrivial, and since $S$ is nonempty, $b$ is nonempty. So we have $\neg(\{\}\vdash^F b)$ by part 4 of Proposition 11. Thus $\{b\}$ is the smallest subset of $B$ that generates $b$. $\square$

3.3. Orthogonality
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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 *$X,Y\in \mathrm{Part}(S)$*, we say *$X$* is orthogonal to *$Y$* (in *$F$*), written *$X \perp^F Y$*, if *$h^F(X)\cap h^F(Y)=\{\}$*.*

*If *$\neg(X \perp^F Y)$*, 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 *$X,Y\in \mathrm{Part}(S)$*,  *$X \perp^F Y$* if and only if there exists a *$C\subseteq B$* such that *$X\leq_S\bigvee_S(C)$* and *$Y\leq_S\bigvee_S(B\setminus C)$*.*

*Proof.* If there exists a $C\subseteq B$ such that $X\leq_S\bigvee_S(C)$ and $Y\leq_S\bigvee_S(B\setminus C)$, then $C\vdash^F X$ and $B\setminus C\vdash^F Y$. Thus, $h^F(X)\subseteq C$ and $h^F(Y)\subseteq B\setminus C$, so $h^F(X)\cap h^F(Y)=\{\}$.

Conversely, if $h^F(X)\cap h^F(Y)=\{\}$, let $C=h^F(X)$. Then $C\vdash^F X$, so $X\leq_S\bigvee_S(C)$, and $B\setminus C\supseteq h^F(Y)$, so $B\setminus C\vdash^F Y$, so $Y\leq_S\bigvee_S(B\setminus C)$. $\square$

Here are some basic properties of orthogonality.

**Proposition 15**. *Let *$F=(S,B)$* be a finite factored set, and let *$X,Y,Z\in \mathrm{Part}(S)$* be partitions of *$S$*.*

1.  *If *$X \perp^F Y$*, then *$Y \perp^F X$*.*
2.  *If *$X \perp^F Z$* and *$Y\leq_S X$*, then *$Y \perp^F Z$*.*
3.  *If *$X \perp^F Z$* and *$Y \perp^F Z$*, then *$(X\vee_S Y) \perp^F Z$*.*
4.  $X \perp^F X$* if and only if *$X=\text{Ind}_S$*.*

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

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

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 *$X,Y\in \mathrm{Part}(S)$*, we say *$X$* is before *$Y$* (in *$F$*), written *$X\leq^F Y$*, if *$h^F(X)\subseteq h^F(Y)$*.*

*We say *$X$* is strictly before *$Y$* (in *$F$*), written *$X<^FY$*, if *$h^F(X)\subset h^F(Y)$*.*

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 *$X,Y\in \mathrm{Part}(S)$*,  *$X\leq^F Y$* if and only if every *$C\subseteq B$* satisfying *$Y\leq_S\bigvee_S(C)$* also satisfies *$X\leq_S\bigvee_S(C)$*.*

*Proof.* Note that by part 7 of Proposition 10, part 5 of Proposition 11, and the definition of history, $C$ satisfies $Y\leq_S\bigvee_S(C)$ if and only if $C\supseteq h^F(Y)$, and similarly for $X$. 

Clearly, if $h^F(Y)\supseteq h^F(X)$, every $C\supseteq h^F(Y)$ satisfies $C\supseteq h^F(X)$. Conversely, if $h^F(X)$ is not a subset of $h^F(Y)$, then we can take $C=h^F(Y)$, and observe that $C\supseteq h^F(Y)$ but not $C\supseteq h^F(X)$. $\square$

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 *$X,Y\in \mathrm{Part}(S)$*, *$X\leq^F Y$* if and only if every *$Z \in \mathrm{Part}(S)$* satisfying *$Y \perp^F Z$* also satisfies *$X \perp^F Z$*.*

*Proof.* Clearly if $h^F(X)\subseteq h^F(Y)$, then every $Z$ satisfying $h^F(Y)\cap h^F(Z)=\{\}$ also satisfies $h^F(X)\cap h^F(Z)=\{\}$.

Conversely, if $h^F(X)$ is not a subset of $h^F(Y)$, let $b\in B$ be an element of $h^F(X)$ that is not in $h^F(Y)$. Assuming $S$ is nonempty, $b$ is nonempty, so we have $h^F(b)=\{b\}$, so $Y \perp^F b$, but not $X \perp^F b$. On the other hand, if $S$ is empty, then $X=Y=\{\}$, so clearly $X\leq^F Y$. $\square$

Here are some basic properties of time.

**Proposition 18**. *Let *$F=(S,B)$* be a finite factored set, and let *$X,Y,Z\in \mathrm{Part}(S)$* be partitions of *$S$*.*

1.  $X\leq^F X$*.*
2.  *If *$X\leq^F Y$* and *$Y\leq^F Z$*, then *$X\leq^F Z$*.*
3.  *If *$X\leq_S Y$*, then *$X\leq^F Y$*.*
4.  *If *$X\leq^F Z$* and *$Y\leq^F Z$*, then *$(X\vee_S Y)\leq^F Z$*.*

*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. $\square$

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 *$X\in \mathrm{Part}(S)$*, *$h^F(X)=\{b\in B\mid b\leq^F X\}$*.*

*Proof.* Since $S$ is nonempty, part 4 of Proposition 13 says that $h^F(b)=\{b\}$ for all $b\in B$. Thus $\{b\in B\mid b\leq^F X\}=\{b\in B\mid \{b\} \subseteq h^F(X)\}=\{b\in B\mid b \in h^F(X)\}=h^F(X)$. $\square$

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