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Revision as of 23:56, 15 April 2009
, 23:56, 15 April 2009→Commentary Note 11.15
<math>\begin{array}{lcccll}
<math>\begin{array}{lcccll}
J & : & \mathbb{R} & \leftarrow & \mathbb{R}
J & : & \mathbb{R} & \leftarrow & \mathbb{R}
& ~\text{(properly restricted)}
& \text{(properly restricted)}
\\[6pt]
\\[6pt]
K & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R}
K & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R}
Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in braces as a name for the 3-adic relation that constitutes or defines the corresponding group, then we have the following set-up:
Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in braces as a name for the 3-adic relation that constitutes or defines the corresponding group, then we have the following set-up:
{| align="center" cellspacing="6" width="90%"
|
: { + } ⊆ '''R''' × '''R''' × '''R'''
<math>\begin{matrix}
J
& : &
[+] \leftarrow [\,\cdot\,]
\\[6pt]
[+]
& \subseteq &
\mathbb{R} \times \mathbb{R} \times \mathbb{R}
\\[6pt]
[\,\cdot\,]
& \subseteq &
\mathbb{R} \times \mathbb{R} \times \mathbb{R}
\end{matrix}</math>
|}
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 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:
