2025/05/21 by Faruk Alpay, Alpay, Faruk · 8 citations
Computer Science · #18B99 #68T27 #F.4.1 #FOS: Mathematics #Formal Methods in Verification #General Mathematics (math.GM) #I.2.3 #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2505.15344
openalex publication_date 2025/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A and define a transfinite evolution functor ϕ\colonA\toA. We prove that the fixed point ϕ∞ 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∞ 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.