LATTICE STRUCTURE OF KNOWLEDGE AND AGREEING TO DISAGREE

This paper is a contribution to the study of the underlying mathematic al structure of common-knowledge, which gives the well-known result of Aumann about the impossibility of 'agreeing to disagree'. We present the Bayesian subjective probability model with player's belief; i.e, a triple (L-i, T-i, mu), in which i is a player, L-i is a lattice in th e field of sets of a stare space Omega, T-i is a correspondence assign ing to each state omega a filter T-i(omega) in L-i and mu is a common- prior. For this model, we impose none of the important restrictions on the information structure in the Aumann-Bacharach model: axiom of kno wledge K-1, axiom of transparency K-2 and axiom of wisdom K-3. We can extend both the disagreement theorem of Aumann and the agreement theor em of Geanacoplos and Polemarchakis under the assumption that each L-i is an Artinian lattice.