Subordination operator
From Groupprops
This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup propertyView a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)
Contents
Definition
Symbol-free definition
The subordination operator is a map from the subgroup property space to itself that sends a subgroup property to the property of being a subgroup for which there exists a ascending chain of subgroups from the subgroup to the group with each member satisfying
in its successor.
Definition with symbols
The subordination operator on a property gives the following property:
satisfies it in
if there is an ascending chain
≤
≤
with each
satisfying
in
.
Property theory of the subordination operator
Transitive and identity-true
As for a general Kleene star operator, the subordination operator is a monotone descendant operator and is also idempotent. The fixed points are precisely the transitive identity-true properties.