Abstract:In this paper we carry on the investigation of partially additive states on quantum logics (see , , , , , , , , etc.). We study a variant of weak states --- the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state --- see ). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of  and presents a rather advanced example in the orthomodular combinatorics (see also , , , , , etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.
Keywords: (weak) state on quantum logic, Greechie paste job, Boolean algebra
AMS Subject Classification: 03G12, 46C05, 81P10