Verb for "swapping" non-commuting operations and modifying them appropriately (commute?)

In mathematics, computer science, physics or any other field that has the concept of commutative operations (or operators), is there a verb to describe the action of taking a sequence AB of two non-commuting operations A and B and modifying each operation so that the inverted sequence B'A' of the modified operations A' and B' has the same effect as AB (i.e. AB = B'A')? Obviously this only makes sense under the constraint that some property of the operations be preserved under the modification, thereby excluding the trivial modification of setting A' = B and B' = A.

I've seen at least one program library refer to this as "commuting" A and B, but I'm not sure if that's correct, given that the usual meaning of "to commute" in this context (cf. Wikipedia article linked above) is that operations can be swapped without making any modifications, e.g. one would say "A and B commute" if AB = BA. On the other hand, perhaps it actually makes perfect sense from that perspective: "If A and B don't commute [on their own], we have to commute them [ourselves, doing something in addition to merely swapping them] if we want to change their order".

I'm just wondering if this is commonly accepted usage for "to commute" or not and whether there are any better expressions if not.

• It depends on the operation. In AB = BA, it's intrinsically multiplication, or whatever passes for it in that group. In "A and B" vs "B and A", it's intrinsically logical conjunction AND. But math is characterized by economizing on operations while multiplying their arguments. Language, on the other hand, multiplies operations and functions that can apply to the comparatively less numerous things/objects/topics that we can identify. So Abelianism pops up here and there (marry is a commutative verb), but it's not normally conserved. Nov 5, 2022 at 17:20
• I'm going with no. When AB commutes to BA, it's the doing of the operator, not the variables. Operators that don't commute do not get to reinvent themselves to commute because someone decides that definition works. The reason I'm shocked. Shocked, I tell you works is that it's a lie both directions. Nov 5, 2022 at 23:36
• Questions about mathematical terminology are better asked in one of the mathematics Stack Exchange sites. You're more likely to find an expert there. Nov 6, 2022 at 17:44
• It depends on whether A' depends only on A, or on both A and B. I can transpose a pair of matrices, but to list equivalent Euler angle tuples, I need to know all of them to derive each element. You are refactoring the (total) operation in the latter case. Nov 7, 2022 at 10:26

In a comment John Lawler wrote:

It depends on the operation. In AB = BA, it's intrinsically multiplication, or whatever passes for it in that group. In "A and B" vs "B and A", it's intrinsically logical conjunction AND. But math is characterized by economizing on operations while multiplying their arguments. Language, on the other hand, multiplies operations and functions that can apply to the comparatively less numerous things/objects/topics that we can identify. So Abelianism pops up here and there (marry is a commutative verb), but it's not normally conserved.

• Could you try to break this down a little in less mathematical language? Are you arguing that commute often implies that the operation is already inherently reversible, but doesn't necessarily mean that? Nov 5, 2022 at 18:52
• It means that there are too many dimensions for something as simple as commutativity. Occasionally you get things that are sort of reversible, on one dimension, but not on all of them. Nov 5, 2022 at 19:40
• @E.T. Here Commute refers to the order of the items being operated upon. A commuter train still goes through my head even now, meaning Back and forth. A times B is the same as B times A. In the briefer form: AB = BA where we infer some operational symbol between the two letters. Apr 6, 2023 at 4:08

Commute is not the best word here, since each field will have its own way of doing things.

In the abstract sense, an operator is applied to an element of a space and returns an element of a space. The two spaces do not have to be identical, although they often are. For example, a typical usage where both spaces are the same would be the partial differentiation operators applied to the space of continuously differentiable functions of two or more variables.

In practice, however, we often want to deal with spaces that are not so nicely behaved, and we introduce additional rules. Sometimes these can be represented in the form of other operators, e.g. projecting a function space onto a smaller and more nicely-behaved space. The meaning of this action typically depends on how the mathematics is being applied. This can vary a great deal.

For example, in rendering a image in computer graphics, suppose we are using a differentiation operator to compute the angle of a surface relative to a ray emanating from a point source of light, in order to compute the brightness of the surface as seen by a virtual camera. However, if our surface has creases, or flat surfaces with sharp joints, it may not be differentiable in all directions at all locations. In such a case we would not be able to differentiate first with respect to one direction, and then with respect to the other (we wouldn’t actually do this as a product of operators for static ray tracing, but it’s hard to think of a good simplified example).

We could, however, find some special way to compute the change in brightness at an edge, possibly by modifying the operators so that they worked locally in a coordinate system that was oriented relative to the edge itself, e.g. in parallel with and perpendicular to the edge. The surface would still be defined by a function, but the domain of this function would be restricted to the points where it could be differentiated in both directions. The edges would be excluded.

This is obviously a highly-specialized application, and the people who work in this field will have their own name for the process.