Non math expert question here.
To keep things more defined let's restrict the Euler angles to Yaw Pitch Roll in that order with all angles between 0 and 2pi.
So there are two YPR's according to the definition above that map to a rotational orientation as well as two unit Quaternions that do the same. For quaternions the unit negative quaternion shares an equivalent rotational orientation with it's positive partner when mapped to 3D space. For euler angles there are two paths that can be taken within 0-2pi that can produce the same rotational orientation, one path is short and straightforward the other path involves rotating 180 degrees across the yaw and flipping back past 90 degrees to achieve the same orientation. Thus when mapping quaternions to euler angles you get two sets with each element of the set mapping to two other elements in the opposing set.
A friend of mine says that there is a technique you can do which is called "isomorphism." (never heard of a technique referred to as an isomorphism would also like to confirm if this is used by mathematicians) Which is to arbitrarily choose that the negative quaternion is mapped to the longer YPR and the positive quaternion is mapped to the shorter YPR and to ignore the other mappings and you can therefore say there is an "isomorphism" between quaternions and euler angles.
I agree that you can choose to ignore certain mappings and create arbitrary mappings such that an isomorphism exists between quaternions and euler angles but is this really how the terminology is used? Do people call this technique an isomorphism or is my friend getting a bit creative here? If I were to approach an expert mathematician and ask him without prior context other then the definition of YPR above if an isomorphism between quaternions and euler angles exist what would he say?
My intuition is telling me if you mess with the mappings between quaternions and euler angles you sort of get something different that's no longer quite a quaternion. It seems to me that is becomes sort of a special quaternion that holds an arbitrary definition that says the negative quaternion maps to a specific YPR. Is my intuition correct or is my friend right?