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| ''P'' : ''X'' ← ''Y'' and ''Q'' : ''Y'' ← ''Z'', we need to determine whether the relational composition ''P'' o ''Q'' ⊆ ''X'' × ''Z'' is also ''P'' o ''Q'' : ''X'' ← ''Z'', or not. | | ''P'' : ''X'' ← ''Y'' and ''Q'' : ''Y'' ← ''Z'', we need to determine whether the relational composition ''P'' o ''Q'' ⊆ ''X'' × ''Z'' is also ''P'' o ''Q'' : ''X'' ← ''Z'', or not. |
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− | <pre>
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| It always helps to begin by recalling the pertinent definitions. | | It always helps to begin by recalling the pertinent definitions. |
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− | For a 2-adic relation L c X x Y, we have: | + | For a 2-adic relation ''L'' ⊆ ''X'' × ''Y'', we have: |
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− | L is a "function" L : X <- Y iff L is 1-regular at Y. | + | : ''L'' is a "function" ''L'' : ''X'' ← ''Y'' if and only if ''L'' is 1-regular at ''Y''. |
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− | As for the definition of relational composition, | + | As for the definition of relational composition, it is enough to consider the coefficient of the composite on an arbitrary ordered pair like ''i'':''j''. |
− | it is enough to consider the coefficient of the | |
− | composite on an arbitrary ordered pair like i:j. | |
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− | (P o Q)_ij = Sum_k (P_ik Q_kj). | + | : (''P'' o ''Q'')<sub>''ij''</sub> = ∑<sub>''k''</sub> (''P''<sub>''ik''</sub> ''Q''<sub>''kj''</sub>). |
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| So let us begin. | | So let us begin. |
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− | P : X <- Y, or P being 1-regular at Y, means that there | + | : ''P'' : ''X'' ← ''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''. |
− | is exactly one ordered pair i:k in P for each k in Y. | |
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− | Q : Y <- Z, or Q being 1-regular at Z, means that there | + | : ''Q'' : ''Y'' ← ''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''. |
− | is exactly one ordered pair k:j in Q for each j in Z. | |
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− | Thus, there is exactly one ordered pair i:j in P o Q | + | Thus, there is exactly one ordered pair ''i'':''j'' in ''P'' o ''Q'' for each ''j'' in ''Z'', which means that ''P'' o ''Q'' is 1-regular at ''Z'', and so we have the function ''P'' o ''Q'' : ''X'' ← ''Z''. |
− | for each j in Z, which means that P o Q is 1-regular | |
− | at Z, and so we have the function P o Q : X <- Z. | |
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| And we are done. | | And we are done. |
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| Bur proofs after midnight must be checked the next day. | | Bur proofs after midnight must be checked the next day. |
− | </pre>
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| ===Commentary Note 11.13=== | | ===Commentary Note 11.13=== |