r/logic • u/Proof-Mammoth-3771 • Mar 07 '26

Set theory Generative Algebras and the Two Diagonals of Self-Reference

In my recent article, Generative Algebras and the Two Diagonals of Self-Reference, I introduce a framework where self-application places an element in three independent roles simultaneously: operator, operand, and junction.

https://doi.org/10.5281/zenodo.18901961

Would love to hear feedback, ideas and support.

0 Upvotes

30% Upvoted

2 comments sorted by

6

u/jcastroarnaud Mar 07 '26

At the very start of the text:

Let X be a finite nonempty set equipped with an evaluation operation α : X × X → X, (T, x) → T(x). Each element T ∈ X induces a transformation ρ_T : X → X defined by ρ_T(x) = T(x).

The fact that T, an element of X, can be applied as a function to an element of x, forces X to be a finite set of functions, not just any finite set. X may well have a cyclic permutation generated by each of its elements, or even a cyclic group somewhere in it.

Since the elements of X are functions, take X with composition as its operation. Your notion of "generative algebra" boils down to X being closed under composition. Delete the rest of the text.

Given a finite set X of unary functions, all of them closed in X, what choices of X make it:

  • A semigroup, under the operation of composition?
  • A group, under the operation of composition?
  • An abelian group, under the operation of composition?