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).