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Revision as of 19:38, 26 November 2007
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===Commentary Note 11.7===
===Commentary Note 11.7===
We come now to the very special cases of 2-adic relations that are known as functions. It will serve a dual purpose on behalf of the exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the RIG material.
We come now to the very special cases of 2-adic relations that are
known as functions. It will serve a dual purpose on behalf of the
exposition if we take the class of functions as a source of object
examples to clarify the more abstruse concepts in the RIG material.
To begin, let's recall the definition of a local flag:
To begin, let's recall the definition of a local flag:
<pre>
L_x@j = {<x_1, ..., x_j, ..., x_k> in L : x_j = x}.
L_x@j = {<x_1, ..., x_j, ..., x_k> in L : x_j = x}.
</pre>
In the case of a 2-adic relation L c X_1 x X_2 = X x Y,
In the case of a 2-adic relation L c X_1 x X_2 = X x Y, we can reap the benefits of a radical simplification in the definitions of the local flags. Also in this case, we tend to denote L_u@1 by "L_u@X" and L_v@2 by "L_v@Y".
we can reap the benefits of a radical simplification in
the definitions of the local flags. Also in this case,
we tend to denote L_u@1 by "L_u@X" and L_v@2 by "L_v@Y".
In the light of these considerations, the local flags of
In the light of these considerations, the local flags of a 2-adic relation L c X x Y may be formulated as follows:
a 2-adic relation L c X x Y may be formulated as follows:
<pre>
L_u@X = {<x, y> in L : x = u}
L_u@X = {<x, y> in L : x = u}
= the set of all ordered pairs in L incident with v in Y.
= the set of all ordered pairs in L incident with v in Y.
</pre>
A sufficient illustration is supplied by the earlier example E.
A sufficient illustration is supplied by the earlier example ''E''.
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
The local flag E_3@X is displayed here:
The local flag E_3@X is displayed here:
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
The local flag E_2@Y is displayed here:
The local flag E_2@Y is displayed here:
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
