 More generally, the sequential product \(a \odot b := \sqrt{a}b\sqrt{a}\) of two effects is another effect, representing the result of observing first \(a\) and then \(b\) in a sequential measurement (and assuming the state updates according to \(W \mapsto (\textrm{Tr}(Wa))^{-1} \sqrt{a} W \sqrt{a}\) after measurement of \(a\)).  Abstracting from this example, S. Gudder and R. J. Greechie ([2002]) defined a sequential effect algebra to be an effect algebra \((\mathbf{L},\oplus,0,1)\) equipped with a binary operation \(\odot : \mathbf{L} \times \mathbf{L} \rightarrow \mathbf{L}\) satisfying the following conditions for all \(a,b,c \in \mathbf{L}\), where \(a | b\) means \(a \odot b = b \odot a\): 