Reichenbachian common cause systems

A partition Ci i∈ I of a Boolean algebra S in a probability measure space (S,p) is called a Reichenbachian common cause system for the correlated pair A,B of events in S if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in (S,p) , and given any finite size n>2, the probability space (S,p) can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of S contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.