The set of asymptotic numbers A introduced in Refs. [1] and [3] is a system of generalized numbers including the system of real numbers R, as well as infinitely small (infinitesimals) and infinitely large numbers. The purpose of this paper is to study in detail the algebraic properties of A which are a little unusual, in a cenain sense, as compared with the known algebraic structures (rings. fields, etc.) This is necessary for the investigation of the class of asymptotic functions [2.4], which are on their part, generalized functions similar to the distributions of Schwartz but allowing the operation of multiplication.

The motivations of this work are in fact, connected with some physical problems [1, 2, 3, 4, 5, 8]; we are going to use the asymptotic numbers and asymptotic functions in the quantum theory in some cases when the multiplication of the distributions is desirable but not possible. Our methods are analogous, in a certain sense, to the methods of the non-standard analysis* [9, 10, 11, 12]. For the sake or convenience we have exposed briefly the most important results of Refs. [1] and [2] and the paper could be read independently of them.



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