Products of elements in vague semigroups and their implementations in vague arithmetic

Vague arithmetic different from the present literature of fuzzy arithmetic has been proposed in [Demirci (Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 10(1) (2002) 25; Internat. J. General Systems 32(2) (2003) 157, 177)] to model vaguely defined arithmetic operations resulting from the indistinguishability of real numbers. The main motivating problem of this paper is to introduce the notion of vague product (sum) of a finite number of real numbers in vague arithmetic, and to point out their fundamental properties. From a more abstract mathematical point of view, the vague product (sum) of a finite number of real numbers in vague arithmetic and their properties can be considered as the vague product of a finite number of elements in vague semigroups and their relevant properties. For this reason, a large part of this paper is devoted to the vague product of a finite number of elements in vague semigroups and their elementary properties. As a direct implementation of the present results, it is shown that the vague product (sum) of a finite number of real numbers in vague arithmetic can be easily evaluated in terms of the underlying many-valued equivalence relations. Furthermore, various non-trivial examples for the vague product (sum) of a finite number of real numbers in vague arithmetic are designed, and a simple technique for the construction of such non-trivial examples is stated.