Sequentially independent effects

A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator $E$ on a Hilbert space that satisfies $0\le E\le I$. We define effects $E_1,E_2,\ldots ,E_n$ to be sequentially independent if the result of any sequential measurement of $E_1,E_2,\ldots,E_n$ does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.