Peirce’s 1870 “Logic of Relatives” • Comment 11.15
I’m going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving mappings, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce’s “number of” function on logical terms.
The structure preserved by a structure-preserving map is just the structure we all know and love as a triadic relation. Very typically, it will be the type of triadic relation that defines the type of binary operation that obeys the rules of a mathematical structure known as a group, that is, a structure satisfying the axioms for closure, associativity, identities, and inverses.
For example, in the case of the logarithm map 
![\begin{array}{lcccll} J & : & \mathbb{R} & \gets & \mathbb{R} & \text{(properly restricted)} \\[6pt] K & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ K(r, s) = r + s \\[6pt] L & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ L(u, v) = u \cdot v \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blcccll%7D++J+%26+%3A+%26+%5Cmathbb%7BR%7D+%26+%5Cgets+%26+%5Cmathbb%7BR%7D+%26+%5Ctext%7B%28properly+restricted%29%7D++%5C%5C%5B6pt%5D++K+%26+%3A+%26+%5Cmathbb%7BR%7D+%26+%5Cgets+%26+%5Cmathbb%7BR%7D+%5Ctimes+%5Cmathbb%7BR%7D+%26+%5Ctext%7Bwhere%7D%7E+K%28r%2C+s%29+%3D+r+%2B+s++%5C%5C%5B6pt%5D++L+%26+%3A+%26+%5Cmathbb%7BR%7D+%26+%5Cgets+%26+%5Cmathbb%7BR%7D+%5Ctimes+%5Cmathbb%7BR%7D+%26+%5Ctext%7Bwhere%7D%7E+L%28u%2C+v%29+%3D+u+%5Ccdot+v++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in brackets as a name for the triadic relation that defines the corresponding group, we have the following set-up.
![\begin{matrix} J & : & [+] \gets [\,\cdot\,] \\[6pt] [+] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \\[6pt] [\,\cdot\,] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++J+%26+%3A+%26+%5B%2B%5D+%5Cgets+%5B%5C%2C%5Ccdot%5C%2C%5D++%5C%5C%5B6pt%5D++%5B%2B%5D+%26+%5Csubseteq+%26+%5Cmathbb%7BR%7D+%5Ctimes+%5Cmathbb%7BR%7D+%5Ctimes+%5Cmathbb%7BR%7D++%5C%5C%5B6pt%5D++%5B%5C%2C%5Ccdot%5C%2C%5D+%26+%5Csubseteq+%26+%5Cmathbb%7BR%7D+%5Ctimes+%5Cmathbb%7BR%7D+%5Ctimes+%5Cmathbb%7BR%7D++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=1&c=20201002)
It often happens that both group operations are indicated by the same sign, usually one from the set 
Figure 50 presents a generic picture for groups 
In a setting where both groups are written with a plus sign, perhaps even constituting the same group, the defining formula of a morphism, 
Resources
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term
cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)
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