Transiter master theorem for composition operator on subgroup properties

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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.