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Commutative DiagramsCommutative diagrams are a particularly nice form of mathematical notation. In case you aren't familiar with them, here's a quick example involving rotations and projections.
Starting from threedimensional Euclidean space in the upper left corner, there are two ways to follow the arrows to the lower right. Going across and then down, we apply a threedimensional rotation followed by a projection; alternatively, going down and then across, we apply the very same projection followed by a twodimensional rotation. We say that the diagram is commutative, or that it commutes, if the two ways of following the arrows always yield the same result … that is, if the following holds for every initial point x.
Now that we know what it means for the diagram to commute, we can ask questions about it. If, for example, we're given a projection and a threedimensional rotation, can we always find a twodimensional rotation that makes the diagram commute? It turns out we can't, not always. The diagram can only be made to commute if the projection is along the axis of rotation.

See AlsoMultiplication in Base 7 Relation to Commutative Diagrams Standard Series, The Synchronization and Merging @ September (2000) 