# Finite-join-closedness is left residual-preserved

From Groupprops

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 View all subgroup metametaproperty dissatisfactions

## Contents

## 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: