Alpay Algebra: A Universal Structural Foundation

• URL: http://arxiv.org/abs/2505.15344v1
• Date: Wed, 21 May 2025 10:18:49 GMT
• Title: Alpay Algebra: A Universal Structural Foundation
• Authors: Faruk Alpay,
• Abstract summary: Alpay Algebra is introduced as a universal, category-theoretic framework.<n>It unifies classical algebraic structures with modern needs in symbolic recursion and explainable AI.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Alpay Algebra is introduced as a universal, category-theoretic framework that unifies classical algebraic structures with modern needs in symbolic recursion and explainable AI. Starting from a minimal list of axioms, we model each algebra as an object in a small cartesian closed category $\mathcal{A}$ and define a transfinite evolution functor $\phi\colon\mathcal{A}\to\mathcal{A}$. We prove that the fixed point $\phi^{\infty}$ exists for every initial object and satisfies an internal universal property that recovers familiar constructs -- limits, colimits, adjunctions -- while extending them to ordinal-indexed folds. A sequence of theorems establishes (i) soundness and conservativity over standard universal algebra, (ii) convergence of $\phi$-iterates under regular cardinals, and (iii) an explanatory correspondence between $\phi^{\infty}$ and minimal sufficient statistics in information-theoretic AI models. We conclude by outlining computational applications: type-safe functional languages, categorical model checking, and signal-level reasoning engines that leverage Alpay Algebra's structural invariants. All proofs are self-contained; no external set-theoretic axioms beyond ZFC are required. This exposition positions Alpay Algebra as a bridge between foundational mathematics and high-impact AI systems, and provides a reference for further work in category theory, transfinite fixed-point analysis, and symbolic computation.

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