## 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 maps, 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 that is preserved by a structure-preserving map is just the structure that 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 that is known as a group, that is, a structure that satisfies the axioms for closure, associativity, identities, and inverses.

For example, in the case of the logarithm map $J$ we have the following data:

$\begin{array}{lcccll} J & : & \mathbb{R} & \gets & \mathbb{R} & \text{(properly restricted)} \\[6pt] K & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ K(r, s) = r + s \\[6pt] L & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ L(u, v) = u \cdot v \end{array}$

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:

$\begin{matrix} J & : & [+] \gets [\,\cdot\,] \\[6pt] [+] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \\[6pt] [\,\cdot\,] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \end{matrix}$

It often happens that both group operations are indicated by the same sign, usually one from the set $\{ \cdot, *, + \}$ 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 one of the following two variants:

$\textit{The image of the sum is the sum of the images.}$

$\textit{The image of the product is the sum of the images.}$

Figure 50 presents a generic picture for groups $G$ and $H.$

 (50)

In a setting where both groups are written with a plus sign, perhaps even constituting the same group, the defining formula of a morphism, $J(L(u, v)) = K(Ju, Jv),$ takes on the shape $J(u + v) = Ju + Jv,$ which looks analogous to the distributive multiplication of a factor $J$ over a sum $(u + v).$ This is why morphisms are regarded as generalizations of linear functions and are frequently referred to in those terms.