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

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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]]. | + | 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>. | + | 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> | + | 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 is a subgroup property. Let denote the left transiter of .

- If is transitive, is stronger than its left transiter. That is, .
- If is identity-true, the left transiter of is stronger than . That is, .
- if is a t.i. subgroup property, i.e., is both transitive and identity-true, then .
- 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: .

### The version for right transiters

Suppose is a subgroup property. Let denote the right transiter of .

- If is transitive, is stronger than its right transiter. That is, .
- If is identity-true, the right transiter of is stronger than . That is, .
- if is a t.i. subgroup property, i.e., is both transitive and identity-true, then .
- 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: . Moreover, the fixed point space of this operator is the same as the fixed point space of the left transiter.