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 acting on a set   Transformation groupies normally refer to 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 where is the identity element and the remaining element satisfies the equation 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 on two variables are listed in Column 1.  Column 2 lists the elements of the set specifically, the sixteen logical graphs 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 into itself are material in this setting.  Column 3 shows the result of the group element acting on each graph in which is of course the same graph back again.  Column 4 shows the result of the group element acting on each graph in which is the entitative graph dual to the existential graph in Column 2.

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 left invariant or unchanged by the respective group actions.  I’ll explain the significance of the fixed point parameter next time.

Resources

• Logic Syllabus
• Logical Graphs
• Duality Indicating Unity
• Futures Of Logical Graphs
• Minimal Negation Operators
• Survey of Theme One Program
• Survey of Animated Logical Graphs
• Propositional Equation Reasoning Systems
• Applications • Constraint Satisfaction Problems

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)