Mathematical Duality in Logical Graphs • Discussion 2

Re: Interpretive Duality in Logical Graphs • 1
Re: Mathematical Duality in Logical Graphs • 1
Re: Laws of Form • Lyle Anderson

LA:

Definition 1. A group $(G, *)$ is a set $G$ together with a binary operation $* : G \times G \to G$ satisfying the following three conditions.

Associativity. For any $x, y, z \in G,$ we have $(x * y) * z = x * (y * z).$
Identity. There is an identity element $e \in G$ such that $\forall g \in G,$
we have $e * g = g * e = g.$
Inverses. Each element has an inverse, that is, for each $g \in G,$
there is some $h \in G$ such that $g * h = h * g = e.$

Dear Lyle,

Thanks for supplying that definition of a mathematical group. It will afford us a wealth of useful concepts and notations as we proceed. As you know, the above three axioms define what is properly called an abstract group. Over the course of group theory’s history that definition was gradually abstracted from the more concrete examples of permutation groups and transformation groups initially arising in the theory of equations and their solvability.

As it happens, the application of group theory I’ll be developing over the next several posts will be using the more concrete type of structure, where a transformation group $G$ is said to “act on” a set $X$ by permuting its elements among themselves. In the work we do here, each group $G$ we contemplate will act a set $X$ which may be viewed as either one of two things, either a canonical set of expressions in a formal language or the mathematical objects denoted by those expressions.

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that’s a ways down the road from here.

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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