Transiter master theorem for composition operator on subgroup properties

Statement

This is the version of the transiter master theorem for the Composition operator (?) on subgroup properties.

The version for left transiters

Suppose $p$ is a Subgroup property (?). Let $L(p)$ denote the left transiter of $p$.

1. If $p$ is transitive, $p$ is stronger than its left transiter. That is, $p \le L(p)$.
2. If $p$ is identity-true, the left transiter of $p$ is stronger than $p$. That is, $L(p) \le p$.
3. if $p$ is a t.i. subgroup property, i.e., $p$ is both transitive and identity-true, then $p = L(p)$.
4. The left transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Together, (3) and (4) tell us that the left transiter operator is idempotent: $L(L(p)) = L(p)$

The version for right transiters

Suppose $p$ is a subgroup property. Let $R(p)$ denote the right transiter of $p$.

1. If $p$ is transitive, $p$ is stronger than its right transiter. That is, $p\le R(p)$.
2. If $p$ is identity-true, the right transiter of $p$ is stronger than $p$. That is, $R(p) \le p$.
3. if $p$ is a t.i. subgroup property, i.e., $p$ is both transitive and identity-true, then $p = R(p)$.
4. The right transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Together, (3) and (4) tell us that the right transiter operator is idempotent: $R(R(p)) = R(p)$. Moreover, the fixed point space of this operator is the same as the fixed point space of the left transiter.