Left tightness theorem

For function restriction expressions for subgroup properties
Suppose $$a$$ and $$b$$ are properties of functions from a group to itself, and $$p$$ is a subgroup property with the function restriction expression:

$$a \to b$$.

In other words, a subgroup $$H$$ satisfies property $$p$$ in a group $$G$$ if and only if every function from $$G$$ to itself satisfying property $$a$$ in $$G$$, restricts to a function from $$H$$ to itself satisfying property $$b$$ in $$H$$.

Then:


 * If $$a \to b$$ is a left tight function restriction expression, i.e., if $$a$$ cannot be weakened further without changing $$b$$, and if $$p$$ is an identity-true subgroup property, i.e., every group has property $$p$$ as a subgroup of itself, then the right transiter for $$p$$ is the subgroup property $$a \to a$$.
 * Otherwise, if $$p$$ is an identity-true subgroup property, let $$c \to b$$ be the left tightening of $$a \to b$$. Then, the right transiter of $$p$$ is the property $$c \to c$$.

Related facts

 * Balanced implies transitive
 * Right tightness theorem