Products of elements and integral powers of elements in vague groups have a significant concern for the development and the applications of vague algebra. Formulation of properties of these notions basically depends on a many-valued Counterpart to the generalized associative law (called the generalized vague associative law in vague groups). For this reason, the present paper and the forthcoming paper [M. Demirci, The generalized associative law in vague groups and its applications-II, Information Sciences, Submitted for publication] are devoted to the formulation of the generalized vague associative law and its applications in vague groups. The generalized vague associative law in vague semigroups, which is the main contribution of this exposition, and some elementary properties of the notion of a product of a finite number of elements in vague semigroups, which cover necessary preparatory results for Part II, are the subjects of this paper. Since the present paper forms an abstract foundation of the product and sum of a finite number of real numbers in Vague arithmetic, some practical applications of the notions of product and sum of a finite number of real numbers in vague arithmetic are also given. (c) 2005 Elsevier Inc. All rights reserved.