An orthomodular lattice admitting no group-valued measure

Abstract

We construct a finite orthomodular lattice <italic>L</italic> such that, for each commutative group <italic>G</italic>, there is no nontrivial <italic>G</italic>-valued measure on <italic>L</italic>. This result extends a result of R. J. Greechie (<italic>Orthogonal lattices admitting no states</italic>, J. Combin. Theory Ser. A <bold>10</bold> (1971), 119-132), and also sheds light on recent investigations in the noncommutative measure theory.