Hi, yes, I made it non-conmutative to make it a little bit more difficult, I think you don't get to see a non conmutative algebra in high school, it was a way to take her out of her comfort area.

However you made a sligth mistake with the shifting part:

"The set part of the new object will be the union of the two sets with all the letters shifted as many places as specified by the integer part of the second TIAM object."

I think you are shifting a little bit too much.