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| ==Statement== | | ==Statement== |
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| ===The version for left transiters=== | | ===General version=== |
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| Suppose <math>p</math> is a subgroup property. Let <math>L(p)</math> denote the [[left transiter]] of <math>p</math>.
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| # If <math>p</math> is [[transitive subgroup property|transitive]], <math>p</math> is stronger than its left transiter. That is, <math>p</math> \le L(p)</math>.
| | ===Special cases=== |
| # If <math>p</math> is [[identity-true subgroup property|identity-true]], the left transiter of <math>p</math> is stronger than <math>p</math>. That is, <math>L(p) \le p</math>.
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| # if <math>p</math> is a [[t.i. subgroup property]], i.e., <math>p</math> is both transitive and identity-true, then <math>p = 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.
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| Together, (3) and (4) tell us that the left transiter operator is idempotent: <math>L(L(p)) = L(p)</math>
| | Special cases of the transiter master theorem include: |
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| ===The version for right transiters===
| | * [[Transiter master theorem for composition operator on subgroup properties]] |
| | | * [[Transiter master theorem for intersection operator on subgroup properties]] |
| Suppose <math>p</math> is a subgroup property. Let <math>R(p)</math> denote the [[right transiter]] of <math>p</math>.
| | * [[Transiter master theorem for join operator on subgroup properties]] |
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| # If <math>p</math> is [[transitive subgroup property|transitive]], <math>p</math> is stronger than its right transiter. That is, <math>p</math> \le R(p)</math>.
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| # If <math>p</math> is [[identity-true subgroup property|identity-true]], the right transiter of <math>p</math> is stronger than <math>p</math>. That is, <math>R(p) \le p</math>.
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| # if <math>p</math> is a [[t.i. subgroup property]], i.e., <math>p</math> is both transitive and identity-true, then <math>p = R(p)</math>.
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| # The right transiter of any property is a [[t.i. subgroup property]], that is, it is both transitive and identity-true.
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| Together, (3) and (4) tell us that the left transiter operator is idempotent: <math>L(L(p)) = L(p)</math>. Moreover, the fixed point space of this operator is the same as the fixed point space of the left transiter.
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| ==Related facts== | | ==Related facts== |
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| * [[Residuation master theorem]] | | * [[Residuation master theorem]] |