Yes, that is what a trademark is for, in general. In this case, the court ruled that the term "open AI" is too generic to qualify for that protection exactly be

Yes, that is what a trademark is for, in general. In this case, the court ruled that the term "open AI" is too generic to qualify for that protection exactly because it is a purely descriptive term that could legitimately refer to any "freely available" model in common parlance.