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 circle plus-idempotency of the circle plus-unit element u results to the idempotency of the pseudo-addition circle plus, i.e., circle plus = v (sup). Several other properties of circle plus and circle plus and their consequences are discussed and illustrated by examples. (C) 2003 Elsevier Inc. All rights reserved.