MATH.22:5.1 - Drop commutativity while preserving invertible composition

Suppose a collection has an associative operation, an identity e and an inverse for every element. These are the group assumptions. Add commutativity, xy=yx, and familiar rearrangements become available. For example, MATH.19’s argument proves (xy)^n=x^n y^n.

The intended new use composes invertible operations whose order matters. Remove commutativity and retain the group assumptions.

Cancellation survives, although its proof may change. A commutative proof multiplies ax=ay on the right by the inverse of a, then rearranges axa⁻¹ and aya⁻¹ to obtain x=y. This uses commutativity. A replacement proof multiplies on the left: a⁻¹(ax)=a⁻¹(ay). Associativity gives (a⁻¹a)x=(a⁻¹a)y, hence x=y by the inverse and identity laws. The theorem therefore survives after removing commutativity.

Unrestricted rearrangement fails. Consider all permutations of {1,2,3}, composed with the rightmost permutation acting first. Composition is associative, the identity leaves each element fixed, and each bijection has an inverse. Let f swap 1 and 2, and let g swap 2 and 3. Then fg sends 1 to 2 to 3 to 1 as a cycle, while f² and g² are identities. Thus (fg)² sends 1 to 3, but f²g² sends 1 to 1.

This is a model of the group assumptions with noncommuting elements. The two-element group {e,h}, with h²=e, is a model where every pair commutes. Under ordinary classical group logic, the two models show that commutativity is independent of the retained group axioms.

The revised theory now admits ordered compositions. It retains cancellation and inverses. Rearrangement remains usable for particular pairs whose commutation is established; it has ceased to be a general permission.