Transformations of Logical Graphs • 1

Posted on May 5, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 1
Re: Mathematical Duality in Logical Graphs • 1

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{Interpretive Duality as Group Symmetry}

Interpretive 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: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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