Commutativity

Commutativity is the property of a binary operation whereby the order of operands does not affect the result: a ⊗ b = b ⊗ a. It is distinct from associativity, which concerns the grouping of operands rather than their order. Commutativity is the formal expression of symmetry in algebraic operations: addition and multiplication of numbers are commutative, but subtraction, division, and matrix multiplication are not.

In physics, commutativity is deeply connected to conservation laws through Noether's theorem: symmetries of physical laws correspond to conserved quantities, and the failure of commutativity — as in the non-commutativity of quantum mechanical operators — signals the absence of a corresponding classical conservation law. In computer science, commutativity determines whether parallel operations on shared data can be reordered without changing the outcome, making it a foundational property for the design of concurrent systems and distributed algorithms.