ONT Re: Logic Of Relatives
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LOR. Note 54
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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 couple of 2-adic relations is again a 2-adic relation,
we know that the relational composition of a couple of 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 of 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.
So our task is this: Given a couple of 2-adic relations,
P c X x Y and Q c Y x Z, that are functional on correlates,
P : X <- Y and Q : Y <- Z, we need to determine whether the
relational composition P o Q c X x Z is also P o Q : X <- Z,
or not.
It always helps to begin by recalling the pertinent definitions:
For a 2-adic relation L c X x Y, we have:
L is a "function" L : X <- Y iff L is 1-regular at Y.
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.
(P o Q)_ij = Sum_k (P_ik Q_kj).
So let us begin.
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.
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.
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.
And we are done.
Bur proofs after midnight must be checked the next day.
Jon Awbrey
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