Peirce’s 1870 “Logic Of Relatives” • Comment 11.9

Among the variety of regularities affecting dyadic relations we pay special attention to the $c$ -regularity conditions where $c$ is equal to $1.$

Let $P \subseteq X \times Y$ be an arbitrary dyadic relation. The following properties of $P$ can then be defined:

$\begin{array}{lll} P ~\text{is total at}~ X & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ X. \\[6pt] P ~\text{is total at}~ Y & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ Y. \\[6pt] P ~\text{is tubular at}~ X & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ X. \\[6pt] P ~\text{is tubular at}~ Y & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ Y. \end{array}$

We previously examined dyadic relations that separately exemplified each of these regularity conditions. And we introduced a few bits of terminology and special-purpose notations for working with tubular relations:

$\begin{array}{lll} P ~\text{is a pre-function}~ P : X \rightharpoonup Y & \iff & P ~\text{is tubular at}~ X. \\[6pt] P ~\text{is a pre-function}~ P : X \leftharpoonup Y & \iff & P ~\text{is tubular at}~ Y. \end{array}$

We arrive by way of this winding stair at the special stamps of dyadic relations $P \subseteq X \times Y$ that are variously described as $1$ -regular, total and tubular, or total prefunctions on specified domains, either $X$ or $Y$ or both, and that are more often celebrated as functions on those domains.

If $P$ is a pre-function $P : X \rightharpoonup Y$ that happens to be total at $X,$ then $P$ is known as a function from $X$ to $Y,$ typically indicated as $P : X \to Y.$

To say that a relation $P \subseteq X \times Y$ is total and tubular at $X$ is to say that $P$ is $1$ -regular at $X.$ Thus, we may formalize the following definitions:

$\begin{array}{lll} P ~\text{is a function}~ P : X \to Y & \iff & P ~\text{is}~ 1\text{-regular at}~ X. \\[6pt] P ~\text{is a function}~ P : X \leftarrow Y & \iff & P ~\text{is}~ 1\text{-regular at}~ Y. \end{array}$

For example, let $X = Y = \{ 0, \ldots, 9 \}$ and let $F \subseteq X \times Y$ be the dyadic relation depicted in the bigraph below:

(39)

We observe that $F$ is a function at $Y$ and we record this fact in either of the manners $F : X \leftarrow Y$ or $F : Y \to X.$

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