Balanced subgroup property (function restriction formalism)

Symbol-free definition
A subgroup property is said to be a balanced subgroup property if it can be expressed via a function restriction expression with both the left side and the right side being equal.

Definition with symbols
A subgroup property is said to be a balanced subgroup property if it can be expressed as $$a \to a$$ where $$a$$ is a function property. In other words, a subgroup $$H$$ satisfies the property in a group $$G$$ if and only if every function on $$G$$ satisfying property $$a$$ in $$G$$ restricts to a function satisfying property $$a$$ on $$H$$.

Characteristic subgroup
The property of a subgroup being characteristic is expressible as a balanaced subgroup property in the function restriction formalism as follows:

Automorphism $$\to$$ Automorphism

Other examples

 * Fully characteristic subgroup = Endomorphism $$\to$$ endomorphism
 * Injective endomorphism-invariant subgroup = Injective endomorphism $$\to$$ injective endomorphism
 * Retraction-invariant subgroup = Retraction $$\to$$ Retraction
 * Transitively normal subgroup = Normal automorphism $$\to$$ Normal automorphism
 * Conjugacy-closed normal subgroup = Class automorphism $$\to$$ Class automorphism
 * Central factor = Inner automorphism $$\to$$ inner automorphism

T.i. subgroup properties
Clearly, any balanced subgroup property with respect to the function restriction formalism is both transitive and identity-true. Hence, it is a t.i. subgroup property.

Interestingly, a partial converse holds by the balance theorem: every t.i. subgroup property that can be expressed using the function restriction formalism, is actually a balanced subgroup property. In fact, more strongly, a balanced expression for the property can be obtained by using either the right tightening operator or the left tightening operator to any starting expression.

Intersection-closedness
In general, a balanced subgroup property need not be intersection-closed.

Join-closedness
In general, a balanced subgroup property need not be join-closed.