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Revision as of 12:18, 13 December 2007
, 12:18, 13 December 2007→Commentary Note 11.12: markup
''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.
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 L c X x Y, we have:
For a 2-adic relation ''L'' ⊆ ''X'' × ''Y'', we have:
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''.
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.
(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>).
So let us begin.
So let us begin.
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.
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.
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.
And we are done.
And we are done.
Bur proofs after midnight must be checked the next day.
Bur proofs after midnight must be checked the next day.
===Commentary Note 11.13===
===Commentary Note 11.13===
