Changes

Line 4,407: Line 4,407:  
<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}
Line 4,419: Line 4,419:  
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:
   −
: ''J'' : { + } &larr; { <math>\cdot</math> }
+
{| align="center" cellspacing="6" width="90%"
 
+
|
: { + } &sube; '''R''' &times; '''R''' &times; '''R'''
+
<math>\begin{matrix}
 
+
J
: { <math>\cdot</math> } &sube; '''R''' &times; '''R''' &times; '''R'''
+
& : &
 +
[+] \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:
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