Alpay Algebra: A Universal Structural Foundation
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arXiv
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| Format: | Preprint |
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2025
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| author | Alpay, Faruk |
| author_facet | Alpay, Faruk |
| contents | 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 $ϕ\colon\mathcal{A}\to\mathcal{A}$. We prove that the fixed point $ϕ^{\infty}$ 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 $ϕ$-iterates under regular cardinals, and (iii) an explanatory correspondence between $ϕ^{\infty}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Alpay Algebra: A Universal Structural Foundation Alpay, Faruk General Mathematics 18B99, 68T27 F.4.1; I.2.3 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 $ϕ\colon\mathcal{A}\to\mathcal{A}$. We prove that the fixed point $ϕ^{\infty}$ 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 $ϕ$-iterates under regular cardinals, and (iii) an explanatory correspondence between $ϕ^{\infty}$ 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. |
| title | Alpay Algebra: A Universal Structural Foundation |
| topic | General Mathematics 18B99, 68T27 F.4.1; I.2.3 |
| url | https://arxiv.org/abs/2505.15344 |