PROPERTIES OF THE UNIVERSAL AGGREGATOR
Keywords:
universal aggregator, monoid, catamorphism, binary operation, reduction, semirings, integral reductionsAbstract
This paper investigates the fundamental structural properties of the universal mathematical aggregator , 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 [1], [3], [7].
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[1] Toshkov, A., Дефиниране на универсален математически агрегатор. КНК, т.14, 2025г. Бургаски свободен университет. (Дефиниране на универсален математически агрегатор)
[2] MAC LANE, Saunders., Categories for the Working Mathematician.Springer-Verlag, 1998.(Kласически труд, дефиниращ моноиди, катаморфизми и универсални свойства.)
[3] BIRD, Richard; DE MOOR, Oege., Algebra of Programming.Prentice Hall, 1997.(Катаморфизми, редукции, fold оператори — фундаментално за агрегиране.)
[4] GOGUEN, Joseph; BURSTALL, Rod., Institutions: Abstract Model Theory for Specification and Programming.Journal of the ACM, 1984.(Универсални конструкции, абстрактни оператори.),
[5] GOLDBLATT, Robert., Topoi – The Categorial Analysis of Logic.Dover, 2006.(Формални оператори, квантори, структурни свойства.)
[6] KLEENE, Stephen., Mathematical Logic.Wiley, 1967.(Квантори, логически оператори – връзка с A∘ чрез специални моноиди.)
[7] HEBERT, Jean-Paul; SIMMONS, Harold., Semirings and their Applications.Springer, 2009.(Семиринги, редукции, дискретни интеграли — директно свързано.)
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Copyright (c) 2025 Ангел Тошков
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