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

From Groupprops

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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=== |

## 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.