Difference between revisions of "Transiter master theorem for composition operator on subgroup properties"

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(The version for right transiters)
 
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==Statement==
 
==Statement==
  
This is the version of the transiter master theorem for the [[fact about::composition operator]] on [[fact about::subgroup property|subgroup properties]].
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This is the version of the transiter master theorem for the [[fact about::composition operator;1| ]][[composition operator]] on [[fact about::subgroup property|subgroup properties]].
  
 
===The version for left transiters===
 
===The version for left transiters===
  
Suppose <math>p</math> is a [[fact about::subgroup property]]. Let <math>L(p)</math> denote the [[left transiter]] of <math>p</math>.
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Suppose <math>p</math> is a [[fact about::subgroup property;3| ]][[subgroup property]]. Let <math>L(p)</math> denote the [[left transiter]] of <math>p</math>.
  
 
# If <math>p</math> is [[transitive subgroup property|transitive]], <math>p</math> is stronger than its left transiter. That is, <math>p \le L(p)</math>.
 
# If <math>p</math> is [[transitive subgroup property|transitive]], <math>p</math> is stronger than its left transiter. That is, <math>p \le L(p)</math>.
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# The left transiter of any property is a [[t.i. subgroup property]], that is, it is both transitive and identity-true.
 
# 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: <math>L(L(p)) = L(p)</math>
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Together, (3) and (4) tell us that the left transiter operator is idempotent: <math>L(L(p)) = L(p)</math>.
  
 
===The version for right transiters===
 
===The version for right transiters===

Latest revision as of 02:11, 21 August 2021

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.