Uniqueness of Products In Higher Algebraic K-Theory

which induces an equivalence between ( — 1)-connected covers of ER and the Gersten-Wagoner spectrum GWR ([3] and [13]). May [6] has given a similar uniquenes theorem for higher algebraic Ktheories (or, infinite loop space machines) defined on permutative (i. eB, symmetric strict monoidal) categories: given an infinite loop space machine E defined on permutative categories, there exists a natural equivalence of spectra between EU and the spectrum SBU constructed by Segal [9]. In the present article we study the multiplicativity of such natural transformations between higher algebraic K-theories defined on permutative categories, or exact categories, or rings. Here the term ^multiplicativity is used in the following sense,, Let E and E' be functors *% —>&* from permutative categories (or exact categories, or rings) to CW-spectra, and suppose that E (resp. E') functorially associates to each pairing UxV-^W in ^ a pairing EU/\EV-^EW (resp, E'U/\E'V-»E'W) of CW^-spectra. Then a natural transformation f:E-*E' is called multiplicative if the following square commutes