A characterization of Boolean algebras (Q5906321)

In a lecture given in Naples in March '92, Pavel Ptak characterized a Boolean algebra on the basis of the ``number'' of valuations defined on it; more precisely Ptak proved that an orthomodular lattice is Boolean if and only if the set of all \([0,1]\)-valued valuations defined on it is unital. In this paper we establish an analogous characterization of Boolean algebras through the studying of \(p\)-ideals and prime ideals in an orthomodular lattice. Ptak's result is then obtained as a consequence for the class of those probability states whose kernel is a \(p\)-ideal and, hence, in particular for the class of valuations. Finally we present, with a different proof, a result of G. Kalmbach that characterizes, using a homomorphism, any ideal which is the intersection of prime ideals.