ONT Re: Theory Of Relations
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TOR. Note 6
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We have now seen enough of the ordinary set-theoretic,
the "non-relative" operations on relations to get the
general idea, but we haven't yet touched the relative
operations on relations. This is apparently so terra
incognita even for many who speak so interpidly about
the compositions, the decompositions, the productions,
and the reductions of all kinds of relations that I'm
sure that it would come as a shock to them when first
they step off their enfolding maps onto terra firma.
So let us amble onward with freshly opened eyes, as if
seeing our place under the sun for the very first time.
| Example 1 Revisited.
|
| A = i:j + j:k + k:i
|
| B = i:k + j:i + k:j
|
| L = i:i + i:j + i:k + j:i + j:j + j:k + k:i + k:j + k:k
The operation on 2-adic relations that Peirce knew under the
names of "relative multiplication" or "relative product" can
be defined, I say, 'DEFINED' in the following way, where for
the sake of a beginning account I shall give this first time
an informal but perfectly adequate version of the definition.
To compute PQ, in general, where P and Q are 2-adic relations,
simply multiply out the two sums in the ordinary distributive
algebraic way, only subject to the following rule for finding
the product of two elementary relations of shapes a:b and c:d.
| (a:b)(c:d) = (a:d) if b = c,
|
| (a:b)(c:d) = 0 otherwise.
Here 0 may be taken as the empty set {}, or anything that serves as
the "additive identity element", meaning that C + 0 = C |_| {} = C,
where C is any set.
For example, we may compute the relative product AB as follows:
| AB = (i:j + j:k + k:i)(i:k + j:i + k:j)
|
| = (i:j)(i:k) + (i:j)(j:i) + (i:j)(k:j) +
|
| (j:k)(i:k) + (j:k)(j:i) + (j:k)(k:j) +
|
| (k:i)(i:k) + (k:i)(j:i) + (k:i)(k:j)
|
| = 0 + i:i + 0 +
| 0 + 0 + j:j +
| k:k + 0 + 0
|
| = i:i + j:j + k:k
|
| = I
You will notice that the very definition of the
relative product of 2-adic relations determines
that the result is again a 2-adic relation.
Therefore, in particular, no 3-adic relation can
result as a relative product of 2-adic relations.
This is all that one means by saying that 3-adic relations
are "irreducible" or "indecomposable" to 2-adic relations,
and it is a matter of basic definition, requiring no proof.
Jon Awbrey
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