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Revision as of 02:06, 16 April 2009
, 02:06, 16 April 2009→Commentary Note 11.15
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In many cases, one finds that both groups are written with the same sign of operation, typically "<math>\cdot</math>", "+", "*", or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like these two variants:
In many cases, one finds that both group operations are indicated by the same sign, typically <math>\cdot\!</math> , <math>*\!</math> , <math>+\!</math> , or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like these two variants:
{| align="center" cellspacing="6" width="90%"
| <p>The image of the sum is the sum of the images.</p>
|-
| <p>The image of the product is the sum of the images.</p>
|}
Figure 22 presents a generic picture for groups <math>G\!</math> and <math>H.\!</math>
Figure 22 presents a generic picture for groups ''G'' and ''H''.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| G H |
| G H |
| @ @ |
| O O |
| /|\ /|\ |
| /|\ /|\ |
| / | \ / | \ |
| / | \ / | \ |
| \ \ \ / |
| \ \ \ / |
| \ / \ / \ / |
| \ / \ / \ / |
| @ @ @ |
| O O O |
| J J J |
| J J J |
| |
| |
Figure 22. Group Homomorphism J : G <- H
Figure 22. Group Homomorphism J : G <- H
</pre>
</pre>
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In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, ''J''(''L''(''u'', ''v'')) = ''K''(''Ju'', ''Jv''), takes on the shape ''J''(''u'' + ''v'') = ''Ju'' + ''Jv'', which looks very analogous to the'distributive multiplication of a sum (''u'' + ''v'') by a factor ''J''. Hence another popular name for a morphism: a "linear" map.
In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, <math>J(L(u, v)) = K(Ju, Jv),\!</math> takes on the shape <math>J(u + v) = Ju + Jv,\!</math> which looks very analogous to the distributive multiplication of a sum <math>(u + v)\!</math> by a factor <math>J.\!</math> Hence another popular name for a morphism: a ''linear'' map.
===Commentary Note 11.16===
===Commentary Note 11.16===
