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

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.

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