We consider some questions
which were brought up in a previous paper: (1) If the product of two proposition
observables is a proposition observable, do the corresponding propositions split?
(2) What is the relationship, if any, between the concepts of compatibility
and simultaneity of proposition observables? It is shown that the answer
to (1) is yes and as a corollary we find the partial answer to (2) that
compatibility implies simultaneity. It is also proved that the sum of two
proposition observables is a proposition observable if and only if the corresponding
propositions are orthogonal and also that (xa∘ xb)n converges weakly to xa∧b as
n →∞.
