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| <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} |
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| 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'' : { + } ← { <math>\cdot</math> }
| + | {| align="center" cellspacing="6" width="90%" |
− | | + | | |
− | : { + } ⊆ '''R''' × '''R''' × '''R''' | + | <math>\begin{matrix} |
− | | + | J |
− | : { <math>\cdot</math> } ⊆ '''R''' × '''R''' × '''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: |