Journal of Mathematical Psychology 42, 393399 (1998) The Hexagon Condition and Additive Representation for Two Dimensions: An Algebraic Approach Edi Karni Johns Hopkins University and Zvi Safra Tel Aviv University Within the algebraic approach the Thomsen condition may be replaced with the hexagon condition to imply the existence of additive representation for two dimensions. In some models the Thomsen condition does not have a natural interpretation whereas the hexagon condition does, which makes it better suited for axiomatic theories.  1998 Academic Press 1. INTRODUCTION Additive representations of binary relations for two dimensional product spaces have been obtained within the algebraic approach using the Thomsen condition (Thomsen (1927) and Blaschke (1928)) and using the condition of triple cancela- tion by Krantz, Luce, Suppes, and Tversky (1971). In the topological approach additive representation for two dimensions using the hexagon condition is given in Debreu (1960) for connected separable topological spaces. Wakker (1989) obtained the same conclusion without the assumption of topological separability. None of these results implies another. In this paper we show that the Thomsen condition may be replaced with the hexagon condition in the algebraic framework of Krantz et al. (1971) to obtain additive representation of preferences for two dimensions. The result of the present paper generalizes the aforementioned results thus providing a unified framework. Article No. MP981204 393 0022-249698 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved. This paper was written during a visit of the second author to Johns Hopkins University. We are grate- ful to David Schmeidler, to an anonymous reviewer, and especially to Peter Wakker for their useful comments and suggestions. Reprint requests should be addressed to Edi Karni, Department of Economics, Johns Hopkins University, Baltimore, MD 21218-2685.