noncommutable

Noncommutable describes a relationship between two or more operations or elements for which order of application affects the result. In mathematics the term is used to indicate that the binary operation between elements does not satisfy the commutative law (a · b = b · a), so swapping operands yields a different outcome. The more common synonym in formal contexts is noncommutative.

Common examples arise in linear algebra and operator theory: matrices A and B are noncommutable when AB ≠ BA, and linear operators on a vector space that do not commute cannot, in general, be simultaneously diagonalized. In quantum mechanics noncommutable observables have a central role; for instance the position and momentum operators x and p satisfy [x, p] = xp − px = iħ, and this noncommutability underlies the Heisenberg uncertainty principle.

Noncommutability has structural consequences for algebraic systems, affecting representation theory, spectral theory, and the classification of algebras. In applied fields it appears whenever sequence matters—examples include function composition, certain database operations, and concurrent computing, where noncommutable actions can produce race conditions or require synchronization.

Usage notes: many texts prefer the adjective noncommutative, especially in formal algebra, but noncommutable is used informally to emphasize the inability to interchange operations. Related concepts include commutativity, the commutator [A, B], and associative versus nonassociative structures.