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Peirce’s 1870 “Logic of Relatives” • Comment 11.7

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$L_{a @ j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = a \}.$

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$\begin{array}{lllll} u \star L & = & L_{u @ X} & = & L_{u @ 1} \\[6pt] L \star v & = & L_{v @ Y} & = & L_{v @ 2} \end{array}$

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$\begin{array}{lll} u \star L & = & L_{u @ X} \\[6pt] & = & \{ (u, y) \in L \} \\[6pt] & = & \text{all pairs in}~ L \subseteq X \times Y ~\text{with}~ u ~\text{in}~ X. \\[9pt] L \star v & = & L_{v @ Y} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{all pairs in}~ L \subseteq X \times Y ~\text{with}~ v ~\text{in}~ Y. \end{array}$

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$\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ y \in Y : (u, y) \in L \} \\[6pt] & = & \text{all points in}~ Y ~\text{which are}~ L\text{-related to}~ u ~\text{in}~ X. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{all points in}~ X ~\text{which are}~ L\text{-related to}~ v ~\text{in}~ Y. \end{array}$