Eventually operator

Symbol-free definition
The eventually operator is an operator from the group property space to the group APS property space, that takes a group property and returns the property of being an APS of groups for which all but finitely many members satisfy the given group property.

Definition with symbols
The eventually operator is an operator from the group property space to the group APS property space, that given a group property $$p$$ returns the property $$Ev(p)$$, defined as follows:

An APS of groups $$(G,\Phi)$$ satisfies $$Ev(p)$$ if there exists an $$n$$ such that $$G_N$$ satisfies $$Ev(p)$$ for all $$N > n$$.