ACU-closed group property: Difference between revisions

Latest revision as of 17:10, 7 September 2008

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This article defines a group metaproperty: a property that can be evaluated to true/false for any group property
View a complete list of group metaproperties

Definition

Symbol-free definition

A group property is termed ACU-closed if, whenever there is an ascending chain of subgroups in a group, each having the group property, the union of those subgroups also has the property.

Definition with symbols

A group property $p$ is termed ACU-closed if, for any group $G$ , any nonempty totally ordered set $I$ , and any ascending chain $H_{i}$ of subgroups of $G$ indexed by ordinals $i\in I$ such that $H_{i}\leq H_{j}$ for $i<j$ , the subgroup:

$\bigcup _{i\in I}H_{i}$

also satisfies property $p$ .

Relation with other metaproperties

Stronger metaproperties

@@ Line 10: / Line 10: @@
 ===Definition with symbols===
-A [[group property]] <math>p</math> is termed '''ACU-closed''' if, for any group <math>G</math>, any limit ordinal <math>\gamma</math>, and any ascending chain <math>H_\alpha</math> of subgroups of <math>G</math> indexed by ordinals <math>\alpha < \gamma</math>, the subgroup:
+A [[group property]] <math>p</math> is termed '''ACU-closed''' if, for any group <math>G</math>, any nonempty totally ordered set <math>I</math>, and any ascending chain <math>H_i</math> of subgroups of <math>G</math> indexed by ordinals <math>i \in I</math> such that <math>H_i \le H_j</math> for <math>i < j</math>, the subgroup:
-<math>\bigcup_{\alpha < \gamma} H_\alpha</math>
+<math>\bigcup_{i \in I} H_i</math>
 also satisfies property <math>p</math>.
@@ Line 20: / Line 20: @@
 ===Stronger metaproperties===
-* [[Weaker than::Varietal group property]]: If the collection of groups satisfying a particular property forms a variety of groups, then the property is closed under unions ascending chains of subgroups.{{further|[[Varietal implies ACU-closed]]}}
+* [[Weaker than::Varietal group property]]
 * [[Weaker than::Union-closed group property]]
+* [[Weaker than::Directed union-closed group property]]
 * [[Weaker than::Join-closed group property]]