УНИВЕРСАЛЕН АГРЕГАТОР 𝑨 ∘ ФОРМАЛНА ТЕОРИЯ, СВОЙСТВА И ПРИЛОЖЕНИЯ
THE UNIVERSAL AGGREGATOR 𝑨 ∘ FORMAL THEORY, PROPERTIES AND APPLICATIONS

Summary/Abstract: This work presents an formal theory of the universal aggregator 𝐴 ∘, an operator that unifies a broad class of discrete reduction processes: sums, products, extrema, Boolean quantifiers, catamorphisms, semiring reductions, graph evaluations, language operations, compositions of transformations and others. Based on a general scheme of selection, evaluation, and structured combination via a binary operation, the aggregator is introduced as a unifying mechanism that incorporates classical logical foundations [1], category– theoretic constructions [5], algebra of programming [11], institutional model theory [16], and semiring-based algebraic systems [19]. An extended definition 𝐴∘∗ is provided, not as a new operator but as a generalization of the original 𝐴 ∘, incorporating context, dependent evaluations, arbitrary finite structures, and controlled reduction order. Fundamental properties and theorems are established, demonstrating the aggregator’s universality as a general reduction operator over discrete structures. Applications span logic, semantics, graph algorithms, and specialized algebraic domains.

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