Purely operator

Definition
The purely operator takes as input a group property for logicists (viz a property for a group (possibly with additional structures and relations) and outputs the corresponding property for a pure group.

In other words, if $$p$$, is a group property for logicists, a group $$G$$ satisfies purely $$p$$, if $$G$$ satisfies $$p$$ when viewed as a pure group (viz without any additional satructure and relations).