| After completely reducible systems, the notions of solvable and nilpotent systems are discussed, where general identities are considered instead of the usual commutativity. |
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| While commutativity holds for many systems, such as the real or complex numbers, there are other systems, such as the system of n × n matrices or the system of quaternions, in which commutativity of multiplication is invalid. |
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| Once the commutativity restriction for meaningful combinations of quantum propositions or properties is in place the usual logical rules apply together with the intuition that goes along with them. |
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| As one can notice, the same operand role is used for both x and y to preserve commutativity of multiplication and addition. |
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| It is well known that commutativity of matrices is very important in the theory of matrices. |
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| Filippis, On derivations and commutativity in prime rings, Int. |
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