Pseudo-arithmetical operations as a basis for the general measure and integration theory

Authors:
Pietro Benvenuti;Radko Mesiar
Affiliations:
Dipartimento di Metodi e Modelli Maternatici per le Scienze Applicate, Universita degli Studi, "La Sapienza", Via Scarpa 16, 00161 Roma, Italy and Gruppo Nazionale di Fisica Matematica C.N.R., 500 ...;Dept. Mathematics, STU, Radlinského 11, 813 68 Bratislava, Slovakia and ÚTIA AVCR Prague, P.O. Box 18, 182 08 Prague, Czech Republic
Venue:
Information Sciences—Informatics and Computer Science: An International Journal
Year:
2004

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Abstract

Several generalizations of the classical measure and integration theory are based on some generalizations of the standard arithmetical operations. The axiomatic approach to the pseudo-arithmetical operations of pseudo-addition and pseudo-multiplication is discussed. Some of required properties strongly influence the structure of these operations (and consequently the resulting measure and integral generalizations). So, e.g., the ⊕-idempotency of the ⊗-unit element u results to the idempotency of the pseudoaddition ⊕, i.e., ⊕ = v (sup). Several other properties of ⊕ and ⊗ and their consequences are discussed and illustrated by examples.