# Directed union-closed group property

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

### Symbol-free definition

A group property is termed **directed union-closed** if given any directed set of subgroups of the group, each satisfying the property, their union also satisfies the property.

### Definition with symbols

A group property is termed **directed union-closed** if given any group , any nonempty directed set , and a collection of subgroups of such that , such that each satisfies , the union:

also satisfies .

## Relation with other metaproperties

### Stronger metaproperties

- Join-closed group property
- Union-closed group property
- Varietal group property:
`For full proof, refer: Varietal implies directed union-closed`