DEFINITION OF A NEW UNIVERSAL MATHEMATICAL AGGREGATOR
Keywords:
universal aggregator, monoid, binary operation, set-based aggregation, summarizing operatorsAbstract
The paper proposes a new mathematical operator—a universal aggregator—that unifies a wide range of classical operators such as sum, product, extrema, quantifiers, integrals, semiring reductions, and others[3], [7], all of which appear as strict special cases of this operator. The universal aggregator is defined, an appropriate calligraphic notation for its representation is chosen, the method of its use is established, and various examples of its application are demonstrated. A special focus is placed on defining the aggregator so that it operates on sets, not merely as a formal fold over lists, and this is justified in a mathematically precise way[1], [3], [5].
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[1] De Angelis, E.; Fioravanti, F.; Pettorossi, A.; Proietti, M. “Catamorphic Abstractions for Constrained Horn Clause Satisfiability.” Theory and Practice of Logic Programming, vol. 25, no. 1, January 2025, pp. 64–91.
[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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