СВОЙСТВА НА УНИВЕРСАЛНИЯ АГРЕГАТОР 𝑨 ∘
PROPERTIES OF THE UNIVERSAL AGGREGATOR 𝑨 ∘

Summary/Abstract: This paper investigates the fundamental structural properties of the universal mathematical aggregator 𝐴 ∘, introduced in [1], which unifies a wide class of classical operators—sum, product, extrema, logical quantifiers, discretized integrals, semiring reductions, and others—viewed as strict special instances of a single operator defined over an appropriate monoid. We provide an extended formal analysis of the essential algebraic and functional properties of the aggregator, including associativity, identity, commutativity (when applicable), invariance, monotonicity, and decomposability. The necessary conditions are derived for the aggregator to operate naturally over sets rather than merely as a list-based fold. A detailed explanation is given for all parameters, structural components, and operators entering the definition, together with the motivation for examining these properties in view of future applications in analysis, computation theory, optimization, data aggregation, integral operators, and matrix reductions.

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