Finite-join-closedness is left residual-preserved

This article gives the statement, and possibly proof, of a subgroup metaproperty (i.e., Finite-join-closed subgroup property (?)) satisfying a subgroup metametaproperty (i.e., Left residual-preserved subgroup metaproperty (?))
View all subgroup metametaproperty satisfactions

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View all subgroup metametaproperty dissatisfactions

Statement

Property-theoretic statement

The Left residual operator for composition (?) of a finite-join-closed subgroup property by any subgroup property is again finite-join-closed.

Related facts

Similar facts about being left residual-preserved

• Join-closedness is left residual-preserved
• Finite-intersection-closedness is left residual-preserved
• Intersection-closedness is left residual-preserved
• Conjugate-join-closedness is left residual-preserved
• Normalizing join-closedness is left residual-preserved

Similar facts about being right residual-preserved

Some similar facts about right residual-preserved are:

• Upper join-closedness is right residual-preserved
• Intermediate subgroup condition is right residual-preserved