Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Not having commutativity of all elements under its operation.
Etymologies
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Examples

NonAbelian is a mathematical term for a system with "noncommutative" properties.

Or that the entropy difference DS is determined by the noncommutative uncertainty
Can a Really, Really Fast Spacecraft Turn Into A Black Hole?  Universe Today

The uncertainty Dx, or noncommutative uncertainty principle is an entropy determined result with
Can a Really, Really Fast Spacecraft Turn Into A Black Hole?  Universe Today

This defines a type of spacetime uncertainty, or an underlying noncommutative structure to spacetime, for the length propagation to the screen Dy ~ cDt with
Can a Really, Really Fast Spacecraft Turn Into A Black Hole?  Universe Today

This results in not just noncommutative geometry, but nonassociative structure as well.
Can a Really, Really Fast Spacecraft Turn Into A Black Hole?  Universe Today

"If on the fundamental level of physics there is no space and no time, as many physicists think," says Heller, "noncommutative geometry could be a suitable tool to deal with such a situation."

Heller's current work focuses on noncommutative geometry and groupoid theory in mathematics which attempts to remove the problem of an initial cosmological singularity at the origin of the universe.

On the typical serious math paper, say, noncommutative loops over Lie algebra, or real homotopy theory of Kaehler manifolds, or compactification on CalabiYau manifolds, my head just starts to spin.

This would include: the formation and decay of (nonsupersymmetric) black holes; graviton scattering at hyperPlanckian energies; physically continuous transitions from one topology, through a quantum state with no geometric interpretation, to a different topology; states where a submanifold of spacetime has a noncommutative geometry; states with a variety of apparent geometric singularities, where the physics is nonsingular.

Quaternion multiplication and matrix multiplication is also noncommutative.
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