Animated Logical Graphs • 61

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (57) • (58) • (59) • (60)

Anything called a duality is naturally associated with a transformation group of order 2, say a group $G$ acting on a set $X.$ Transformation groupies normally refer to $X$ as a set of “points” even when the elements have additional structure of their own, as they often do. A group of order two has the form $G = \{ 1, t \},$ where $1$ is the identity element and the remaining element $t$ satisfies the equation $t^2 = 1,$ being on that account self-inverse.

A first look at the dual interpretation of logical graphs from a group-theoretic point of view is provided by the following Table. The sixteen boolean functions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ on two variables are listed in Column 1. Column 2 lists the elements of the set $X,$ specifically, the sixteen logical graphs $\gamma$ giving canonical expression to the boolean functions in Column 1. Column 2 shows the graphs in existential order but the order is arbitrary since only the transformations of the set $X$ into itself are material in this setting. Column 3 shows the result $1 \gamma$ of the group element $1$ acting on each graph $\gamma$ in $X,$ which is of course the same graph $\gamma$ back again. Column 4 shows the result $t \gamma$ of the group element $t$ acting on each graph $\gamma$ in $X,$ which is the entitative graph dual to the existential graph in Column 2.

$\text{Peirce Duality as Group Symmetry}$

The last Row of the Table displays a statistic of considerable interest to transformation group theorists. It is the total incidence of fixed points, in other words, the number of points in $X$ left invariant or unchanged by the respective group actions. I’ll explain the significance of the fixed point parameter next time.

Resources

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) • Structural Modeling (1) (2)
• Systems Science (1) (2)

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