Changes

<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:

: ''J'' : { + } &larr; { <math>\cdot</math> }

{| align="center" cellspacing="6" width="90%"

 

: { + } &sube; '''R''' &times; '''R''' &times; '''R'''

<math>\begin{matrix}

 

: { <math>\cdot</math> } &sube; '''R''' &times; '''R''' &times; '''R'''

& : &

[+]

& \subseteq &

\mathbb{R} \times \mathbb{R} \times \mathbb{R}

[\,\cdot\,]

& \subseteq &

\mathbb{R} \times \mathbb{R} \times \mathbb{R}

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: