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===Commentary Note 11.12===

===Commentary Note 11.12===

Since functions are special cases of 2-adic relations, and since the space of 2-adic relations is closed under relational composition — in other words, the composition of a pair of 2-adic relations is again a 2-adic relation — we know that the relational composition of two functions has to be a 2-adic relation.  If it is also necessarily a function, then we would be justified in speaking of ''functional composition', and also in saying that the space of functions is closed under this functional form of composition.

Since functions are special cases of 2-adic relations, and since the space of 2-adic relations is closed under relational composition — in other words, the composition of a pair of 2-adic relations is again a 2-adic relation — we know that the relational composition of two functions has to be a 2-adic relation.  If it is also necessarily a function, then we would be justified in speaking of ''functional composition'', and also in saying that the space of functions is closed under this functional form of composition.

Just for novelty's sake, let's try to prove this for relations that are functional on correlates.

Just for novelty's sake, let's try to prove this for relations that are functional on correlates.

The task is this — We are given a pair of 2-adic relations:

The task is this — We are given a pair of 2-adic relations:

{| align="center" cellspacing="6" style="text-align:center" width="90%"

{| align="center" cellspacing="6" width="90%"

| <math>P \subseteq X \times Y \quad \text{and} \quad Q \subseteq Y \times Z</math>

| <math>P \subseteq X \times Y \quad \text{and} \quad Q \subseteq Y \times Z</math>

|}

|}

<math>P\!</math> and <math>Q\!</math> are assumed to be functional on correlates, a premiss that we express as follows:

<math>P\!</math> and <math>Q\!</math> are assumed to be functional on correlates, a premiss that we express as follows:

{| align="center" cellspacing="6" style="text-align:center" width="90%"

{| align="center" cellspacing="6" width="90%"

| <math>P : X \leftarrow Y \quad \text{and} \quad Q : Y \leftarrow Z</math>

| <math>P : X \leftarrow Y \quad \text{and} \quad Q : Y \leftarrow Z</math>

|}

|}

We are charged with deciding whether the relational composition <math>P \circ Q \subseteq X \times Z</math> is also functional on correlates, that is, whether <math>P \circ Q : X \leftarrow Z</math> or not.

We are charged with deciding whether the relational composition <math>P \circ Q \subseteq X \times Z</math> is also functional on correlates, in symbols, whether <math>P \circ Q : X \leftarrow Z.</math>

It always helps to begin by recalling the pertinent definitions.

It always helps to begin by recalling the pertinent definitions.

For a 2-adic relation <math>L \subseteq X \times Y,</math> we have:

For a 2-adic relation <math>L \subseteq X \times Y,</math> we have:

{| align="center" cellspacing="6" style="text-align:center" width="90%"

{| align="center" cellspacing="6" width="90%"

|

|

<math>\begin{array}{lll}

<math>\begin{array}{lll}

|}

|}

As for the definition of relational composition, it is enough to consider the coefficient of the composite on an arbitrary ordered pair like <math>i:j.\!</math>

As for the definition of relational composition, it is enough to consider the coefficient of the composite relation on an arbitrary ordered pair, <math>i\!:\!j.</math>

{| align="center" cellspacing="6" style="text-align:center" width="90%"

{| align="center" cellspacing="6" width="90%"

| <math>(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}</math>

| <math>(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}</math>

|}

|}

So let us begin.

So let us begin.

: ''P''&nbsp;:&nbsp;''X''&nbsp;&larr;&nbsp;''Y'', or ''P'' being 1-regular at ''Y'', means that there is exactly one ordered pair ''i'':''k'' in ''P'' for each ''k'' in ''Y''.

 

: ''Q''&nbsp;:&nbsp;''Y''&nbsp;&larr;&nbsp;''Z'', or ''Q'' being 1-regular at ''Z'', means that there is exactly one ordered pair ''k'':''j'' in ''Q'' for each ''j'' in ''Z''.

<p><math>P : X \leftarrow Y,</math> or the fact <math>P ~\text{is}~ 1\text{-regular at}~ Y,</math> means that there is exactly one ordered pair <math>i\!:\!k \in P</math> for each <math>k \in Y.</math><p>

 

Thus, there is exactly one ordered pair ''i'':''j'' in ''P''&nbsp;o&nbsp;''Q'' for each ''j'' in ''Z'', which means that ''P''&nbsp;o&nbsp;''Q'' is 1-regular at ''Z'', and so we have the function ''P''&nbsp;o&nbsp;''Q''&nbsp;:&nbsp;''X''&nbsp;&larr;&nbsp;''Z''.

<p><math>Q : Y \leftarrow Z,</math> or the fact that <math>Q ~\text{is}~ 1\text{-regular at}~ Z,</math> means that there is exactly one ordered pair <math>k\!:\!j \in Q</math> for each <math>j \in Z.</math><p>

<p>As a result, there is exactly one ordered pair <math>i\!:\!j \in P \circ Q</math> for each <math>j \in Z,</math> which means that <math>P \circ Q ~\text{is}~ 1\text{-regular at}~ Z,</math> and so we have the function <math>P \circ Q : X \leftarrow Z.</math>

And we are done.

And we are done.