Procedure Defined Units
How come 1 meter is a good unit but 1 tuberculine unit is not?
How come UCUM has both of these units?
How come the equality of 1 Bq and 1 Hz is not a problem, but the equality of 1 tuberculine unit and 1 allergy unit is a problem?
How come glucose measured in 1 gram or 1 mol is not a problem?
How can we have both allergen unit [AU] and biological equivalent allergy unit [BAU] in one system?
How come 1 Gray and 1 Sievert both exist in the SI, are they equal?
This is the biggest challenge that the UCUM project is currently facing.
I think the problem always occurs when we measure one thing by an equivalent of another.
Like if I measured an amount of people by the amount of oxygen they consume in the room.
And it gets worse if the measurand and the equivalent quantity are of the same dimension,
Like if I measure the number of people by the number of hairs they leave behind.
And it is all fine if I have a fixed constant conventional coefficient, Like: a person loses 10 hairs per hour, so if after 2 hours a room has 106 hairs on the floor, there must have been 5*104 people in the room.
But it gets worse if an expert comes along and says:
Wait, you can’t tell how many people are in the room, all you know is how much hair they lost!
At this very point you start measuring amount of people by some procedure defined unit:
Like: HEU (hair equivalent unit) the amount of people in a room who lose 10 million hairs in 1 hours.
At this point people demand that UCUM has the HEU unit.
But the problem is, this definition of HEU is explicitly not directly related to the straight-forward number of people in the room, but it is a new kind of quantity, locked to this idiosyncratic proxy measure.
And that now has established its very own dimension also, because in a dimension all units are comparable in fixed conversion formulas (usually proportional, linear, but at least a conventional a-priori defined function.) But not here, because the whole premise was that people rejected the idea of just relating the unit directly to number of people or just reporting number of people directly.
Thus all procedure defined units are dimensions unto themselves.
And thus our simple 7 component dimension vector stops being a vector of any finite length.
What do we do?
The present approach was to add a new property "arbitrary unit" into the unit definition. This measure at least prevents units to be falsely taken as equivalent. But this also prevents any kind of conversion, even between 1 HEU and 1 kHEU. This is presently the best we can do, and it is a good idea to limit the ability to form complex terms with arbitrary units. It might not even make sense in some cases (e.g., if the unit is not on a ratio scale how can one even multiply it?)
But we need a final solution that deals with this challenge.
One possible solution is to assume a hard-line scientific position and put any and all UCUM units on a black-list that are used to merely decorate dimensionless number, indexes, or ratios. A very good argument for this position can be found in #27 (the strike-through only indicates this issue is closed, not that it is invalid).
A Way Forward
Arbitrary units [IU] and [iU] are equal, they are, in fact, the same (just literal difference). The same issue you have with the case-insensitive and case-sensitive lexical variants. So, if we use the isArbitrary property to force checking the unit, we must not fail equal because of those differences.
Therefore, we probably need to extend our dimension and base-unit vector with additional fields, and when checking dimensionality, we need to take those into account. This makes the dimension and base-unit vector sparse at the end. There are 103 arbitrary units to reckon with.