Arithmetic of complex sets

Abstract

Let\(\mathbb{I}\)(ℝ) be the set of all real closed intervals and letΩ1:= {+, −, ×, /} be the set of arithmetic operators of ℝ. By extendingΩ1 from ℝ to\(\mathbb{I}\)(ℝ) as usual one finds that\(\mathbb{I}\)(ℝ) is closed with respect to the operations fromΩ1 (R. E. Moore [9]). In the literature several possibilities are discussed to go over from complex numbers to “complex intervals”: rectangles (Alefeld [1] et al.), discs (Henrici [4] et al.) or ellipses (Kahan [5] et al.). In all three cases the resulting sets are not closed with respect toΩ1, since the multiplication and division of such “intervals” does not lead to sets of the same kind. In what follows the question is treated whether there are classes of complex sets (“generalized intervals”) which are closed with respect toΩ1 or to subsets ofΩ1. One such class is easy to find. Also the shape of the sets involved is discussed. If it is assumed however that the sets under consideration are described by a finite number of parameters then there isno such class closed underΩ1.